Differentiation of Vectors

Having laid down the algebraic foundations of geometric vectors, we now turn our attention to differentiation. When most people think of differentiation of vectors, they imagine a component-by-component operation, such as U(t)=(U1(t),U2(t),U3(t))\mathbf{U}^{\prime}\left( t\right) =\left( U_{1}^{\prime}\left( t\right) ,U_{2}^{\prime}\left( t\right) ,U_{3}^{\prime}\left( t\right) \right) . This way of thinking is an example of the presumptive association between geometric vectors and their components. It is imperative to break this association and to commit to vectors as geometric objects. As we are about to demonstrate, geometric vectors in a Euclidean space have just enough analytical structure to be meaningfully differentiated. Differentiation of geometric vectors will prove to be a crucial element in our approach to Tensor Calculus.
Recall the standard definition of the derivative U(x)U^{\prime}\left( x\right) of an ordinary function U(x)U\left( x\right) , i.e.
U(x)=limh0U(x+h)U(x)h(4.1)U^{\prime}\left( x\right) =\lim_{h\rightarrow0}\frac{U\left( x+h\right) -U\left( x\right) }{h}\tag{4.1}
What is required of U(x)U\left( x\right) in order for this definition to apply? Clearly, the quantity represented by U(x)U\left( x\right) must be subject to three basic operations: addition, multiplication by numbers, and evaluation of limits. Addition (which implies subtraction) is required in order to evaluate the difference U(x+h)U(x)U\left( x+h\right) -U\left( x\right) . Multiplication by numbers (which implies division) is required in order to divide the difference U(x+h)U(x)U\left( x+h\right) -U\left( x\right) by hh. Finally, the need for limits is self-evident.
We have already established that geometric vectors can be added and multiplied by numbers. Thus, we only need to confirm that geometric vectors are subject to evaluation of limits. What makes limits of geometric vectors possible is the availability of Euclidean length since it enables us to state how close two vectors are by measuring the distance between them. The distance between two vectors U\mathbf{U} and V\mathbf{V}, denoted by d(U,V)\operatorname{d}\left( \mathbf{U,V}\right) , is defined as the length of their difference, i.e.
d(U,V)=len(UV).(4.2)\operatorname{d}\left( \mathbf{U,V}\right) =\operatorname{len}\left( \mathbf{U}-\mathbf{V}\right) .\tag{4.2}
With the distance function in hand, we are able to carry the classical definition of a limit over to geometric vectors.
In formal terms, a vector U\mathbf{U} is the limit of a sequence Un\mathbf{U}_{n} if the distance between Un\mathbf{U}_{n} and U\mathbf{U} approaches 00 as nn approaches infinity, i.e.
U=limnUn    if    limnd(Un,U)=0.(4.3)\mathbf{U}=\lim_{n\rightarrow\infty}\mathbf{U}_{n}\text{\ \ \ \ if\ \ \ \ } \lim_{n\rightarrow\infty}\operatorname{d}\left( \mathbf{U}_{n},\mathbf{U} \right) =0.\tag{4.3}
The following figure illustrates a sequence of vectors Un\mathbf{U}_{n} that approach the "vertical" vector U\mathbf{U}.
(4.4)
The same approach can be used to define the limit of a vector-valued function U(γ)\mathbf{U}\left( \gamma\right) . Specifically, the vector U\mathbf{U} is the limit of U(γ)\mathbf{U}\left( \gamma\right) at γ=γ0\gamma=\gamma_{0} if the distance between U(γ)\mathbf{U}\left( \gamma\right) and U\mathbf{U} approaches 00 as γ\gamma approaches γ0\gamma_{0}, i.e.
U=limγγ0U(γ)    if    limγγ0d(U(γ),U)=0.(4.5)\mathbf{U}=\lim_{\gamma\rightarrow\gamma_{0}}\mathbf{U}\left( \gamma\right) \text{\ \ \ \ if\ \ \ \ }\lim_{\gamma\rightarrow\gamma_{0}}\operatorname{d} \left( \mathbf{U}\left( \gamma\right) ,\mathbf{U}\right) =0.\tag{4.5}
Thus, the concept of a limit for vector-valued sequences and functions is established.
Having established the concept of a limit, we can mimic the classical definition
U(x)=limh0U(x+h)U(x)h(4.1)U^{\prime}\left( x\right) =\lim_{h\rightarrow0}\frac{U\left( x+h\right) -U\left( x\right) }{h} \tag{4.1}
of the derivative to give a formal definition of the derivative U(γ)\mathbf{U} ^{\prime}\left( \gamma\right) of a vector-valued function U(γ)\mathbf{U} \left( \gamma\right) that, i.e.
U(γ)=limh0U(γ+h)U(γ)h.(4.6)\mathbf{U}^{\prime}\left( \gamma\right) =\lim_{h\rightarrow0}\frac {\mathbf{U}\left( \gamma+h\right) -\mathbf{U}\left( \gamma\right) }{h}.\tag{4.6}
We reiterate that all aspects of the procedure encoded in this definition can, at least conceptually, be carried out by pure geometric means without the use of a coordinate system.
The derivative of a vector function corresponds to our intuition for rate of change, except now the rate of change is a vector quantity. Nevertheless, many of the ideas from ordinary Calculus carry over to vector-valued functions. For example, the derivative can be used for linear approximations. Namely, for a small hh, the derivative U(γ)\mathbf{U}^{\prime }\left( \gamma\right) can be approximated by the quotient
U(γ+h)U(γ)h(4.7)\frac{\mathbf{U}\left( \gamma+h\right) -\mathbf{U}\left( \gamma\right) }{h}\tag{4.7}
in the sense that the distance between U(γ)\mathbf{U}^{\prime}\left( \gamma\right) and the above quotient is small. Therefore, U(γ+h)\mathbf{U} \left( \gamma+h\right) can be approximated by the familiar formula
U(γ+h)U(γ)+hU(γ),(4.8)\mathbf{U}\left( \gamma+h\right) \approx\mathbf{U}\left( \gamma\right) +h\mathbf{U}^{\prime}\left( \gamma\right) ,\tag{4.8}
also in the sense that the distance between U(γ+h)\mathbf{U}\left( \gamma+h\right) and U(γ)+hU(γ)\mathbf{U}\left( \gamma\right) +h\mathbf{U}^{\prime }\left( \gamma\right) is small.
A vector-valued function U(γ)\mathbf{U}\left( \gamma\right) has the inescapable geometric interpretation as the curve traced out by the tips of the vectors U(γ)\mathbf{U}\left( \gamma\right) emanating from a single point OO known as the origin. In this context, the vectors U(γ)\mathbf{U}\left( \gamma\right) are referred to as the position vectors for the points on the curve. Since it is common to denote position vectors by the letter R\mathbf{R} rather than U\mathbf{U}, we will now switch to that convention and consider a vector-valued function R(γ)\mathbf{R} \left( \gamma\right) and refer to it as the vector equation of the curve.
(4.9)
To reiterate: there is a natural one-to-one correspondence between vector-valued functions and curves. Any vector-valued function R(γ)\mathbf{R} \left( \gamma\right) can be visualized as a curve with respect to a fixed origin OO. Conversely, every curve represents a vector-valued function R(γ)\mathbf{R}\left( \gamma\right) once a fixed origin OO is chosen and a numerical value of a parameter γ\gamma is assigned to every point on the curve in an appropriate fashion. If the parameter γ\gamma represents time, then the curve associated with the function R(γ)\mathbf{R}\left( \gamma\right) can be interpreted as a trajectory of a moving particle.
Given this interpretation of a vector-valued function R(γ)\mathbf{R}\left( \gamma\right) , what is the meaning of the vector R(γ)\mathbf{R}^{\prime }\left( \gamma\right) ? We will now demonstrate R(γ)\mathbf{R}^{\prime}\left( \gamma\right) is a vector that points in the tangential direction to the curve. We intuitively understand the tangent line to be a straight line that "touches" the curve at a single point, typically without crossing it. Alternatively, we can think of the tangent as the straight line that presents itself when one sufficiently zooms in on the curve. Our present goal is to confirm that the definition of R(γ)\mathbf{R}^{\prime}\left( \gamma\right) is consistent with our intuition for the tangent line. We will do so by examining the limiting procedure encoded in the analytical expression for R(γ)\mathbf{R} ^{\prime}\left( \gamma\right) , i.e.
R(γ)=limh0R(γ+h)R(γ)h.(4.10)\mathbf{R}^{\prime}\left( \gamma\right) =\lim_{h\rightarrow0}\frac {\mathbf{R}\left( \gamma+h\right) -\mathbf{R}\left( \gamma\right) }{h}.\tag{4.10}
Once our intuition is confirmed, we will reverse the logic and define the tangent direction as the direction of the vector R(γ)\mathbf{R}^{\prime }\left( \gamma\right) . Additionally, if R(γ)\mathbf{R}\left( \gamma\right) is interpreted as the trajectory of a particle, where γ\gamma represents time, then the velocity of the particle will be defined to be R(γ)\mathbf{R}^{\prime}\left( \gamma\right) .
The following figure shows R(γ)\mathbf{R}\left( \gamma\right) for two points corresponding to two nearby values γ\gamma and γ+h\gamma+h of the parameter.
(4.11)
The difference R(γ+h)R(γ)\mathbf{R}\left( \gamma+h\right) -\mathbf{R}\left( \gamma\right) is the vector from the tip of R(γ)\mathbf{R}\left( \gamma\right) to the tip of R(γ+h)\mathbf{R}\left( \gamma+h\right) .
(4.12)
Importantly, note that the difference R(γ+h)R(γ)\mathbf{R}\left( \gamma+h\right) -\mathbf{R}\left( \gamma\right) is independent of the location of the origin OO. This is why the location of OO is almost always irrelevant. If γ\gamma is thought of as time, R(γ+h)R(γ)\mathbf{R}\left( \gamma+h\right) -\mathbf{R}\left( \gamma\right) corresponds to the displacement of the particle over the short period of time hh.
Dividing the difference R(γ+h)R(γ)\mathbf{R}\left( \gamma+h\right) -\mathbf{R}\left( \gamma\right) by hh yields our initial approximation to the eventual vector R(γ)\mathbf{R}^{\prime}\left( \gamma\right) .
(4.13)
When γ\gamma is thought of as time, the ratio
R(γ+h)R(γ)h(4.14)\frac{\mathbf{R}\left( \gamma+h\right) -\mathbf{R}\left( \gamma\right) }{h}\tag{4.14}
corresponds to the average velocity of the moving particle over a period of time hh.
Finally, imagine what happens to the vector represented by the above ratio as hh begins to approach zero, and observe that its direction begins to approach what we intuitively understand as the tangent to the curve.
(4.15)
Thus, we have justified the interpretation of the derivative R(γ)\mathbf{R} ^{\prime}\left( \gamma\right) of a vector-valued function R(γ)\mathbf{R} \left( \gamma\right) as the tangent to the curve represented by R(γ)\mathbf{R}\left( \gamma\right) .
Having gained this important insight, let us determine R(γ)\mathbf{R}^{\prime }\left( \gamma\right) for a specific curve that will prove to be of great importance in our future investigations. Suppose that R(γ)\mathbf{R}\left( \gamma\right) traces out the unit circle where the parameter γ\gamma corresponds to the central angle measured in the counterclockwise direction with respect to an arbitrary ray emanating from the center of the circle.
(4.16)
Since, as we remarked previously, R(γ)\mathbf{R}^{\prime}\left( \gamma\right) is independent of the location of the origin OO, let us place it at the center of the circle. We have already established that R(γ)\mathbf{R}^{\prime }\left( \gamma\right) is tangential to the circle and it is evident that it points in the counterclockwise direction. Thus, the only remaining quantity to be determined is its length.
If we think of γ\gamma as time and the unit circle as the trajectory of a material particle corresponding to R(γ)\mathbf{R}\left( \gamma\right) , then it is clear that R(γ)\mathbf{R}^{\prime}\left( \gamma\right) is unit length. Indeed, as γ\gamma changes from 00 to 2π2\pi, the particle makes a single revolution and thus travels a distance of 2π2\pi. Therefore, its speed, i.e. the magnitude of R(γ)\mathbf{R}^{\prime}\left( \gamma\right) , has the constant value of
2π2π=1.(4.17)\frac{2\pi}{2\pi}=1.\tag{4.17}
In other words, R(γ)\mathbf{R}^{\prime}\left( \gamma\right) is a unit tangent vector that points in the counterclockwise direction.
(4.18)
The same conclusion regarding the magnitude of R(γ)\mathbf{R}^{\prime}\left( \gamma\right) can be reached by a more formal calculation. Consider the configuration involving R(γ)\mathbf{R}\left( \gamma\right) and R(γ+h)\mathbf{R} \left( \gamma+h\right) for two nearby values of the parameter.
(4.19)
From the isosceles triangle with the vertices at OO, R(γ)\mathbf{R}\left( \gamma\right) , and R(γ+h)\mathbf{R}\left( \gamma+h\right) , we calculate that
len(R(γ+h)R(γ))=2sin(h/2)(4.20)\operatorname{len}\left( \mathbf{R}\left( \gamma+h\right) -\mathbf{R} \left( \gamma\right) \right) =2\sin\left( h/2\right)\tag{4.20}
and therefore
lenR(γ+h)R(γ)h=sin(h/2)h/2.(4.21)\operatorname{len}\frac{\mathbf{R}\left( \gamma+h\right) -\mathbf{R}\left( \gamma\right) }{h}=\frac{\sin\left( h/2\right) }{h/2}.\tag{4.21}
From ordinary Calculus, we know that
limh0sin(h/2)h/2=1.(4.22)\lim_{h\rightarrow0}\frac{\sin\left( h/2\right) }{h/2}=1.\tag{4.22}
Therefore,
limh0(lenR(γ+h)R(γ)h)=1,(4.23)\lim_{h\rightarrow0}\left( \operatorname{len}\frac{\mathbf{R}\left( \gamma+h\right) -\mathbf{R}\left( \gamma\right) }{h}\right) =1,\tag{4.23}
and thus R(γ)\mathbf{R}^{\prime}\left( \gamma\right) is indeed unit length, consistent with the earlier conclusion based on our kinematic intuition.
Finally, for a circle of radius rr, similarly parameterized by the central angle γ\gamma,
(4.24)
the derivative R(γ)\mathbf{R}^{\prime}\left( \gamma\right) is the tangential vector of length rr, i.e.
lenR(γ)=r(4.25)\operatorname{len}\mathbf{R}^{\prime}\left( \gamma\right) =r\tag{4.25}
Proving this fact is left as an exercise.
Geometric vectors are subject to three operations: addition, multiplication by scalars, and the dot product. Naturally, to each operation there corresponds its own differentiation rule.

4.4.1The sum and product rules

Consider two vector-valued functions U(γ)\mathbf{U}\left( \gamma\right) and V(γ)\mathbf{V}\left( \gamma\right) . The derivative applied to their sum is governed by the familiar sum rule
(U+V)=U+V.(4.26)\left( \mathbf{U}+\mathbf{V}\right) ^{\prime}=\mathbf{U}^{\prime} +\mathbf{V}^{\prime}.\tag{4.26}
For the product of a vector-valued function with a constant scalar, the differentiation rule reads
(cU)=cU.(4.27)\left( c\mathbf{U}\right) ^{\prime}=c\mathbf{U}^{\prime}.\tag{4.27}
If the scalar cc is itself a function of γ\gamma, then the product rule reads
(cU)=cU+cU.(4.28)\left( c\mathbf{U}\right) ^{\prime}=c^{\prime}\mathbf{U}+c\mathbf{U} ^{\prime}.\tag{4.28}
The proofs of these rules are left as exercises. While demonstrating the first two identities is entirely straightforward, the last one may pose a challenge. We recommend applying the same approach that we are about to apply to the dot product rule.

4.4.2The dot product rule

We will now demonstrate that vector-valued functions satisfy the dot product rule
(UV)=UV+UV,(4.29)\left( \mathbf{U}\cdot\mathbf{V}\right) ^{\prime}=\mathbf{U}^{\prime} \cdot\mathbf{V}+\mathbf{U}\cdot\mathbf{V}^{\prime},\tag{4.29}
which is entirely analogous to the product rule in ordinary Calculus. Not surprisingly, we will also be able to demonstrate this rule by borrowing an argument from Calculus.
Let
F(γ)=U(γ)V(γ)(4.30)F\left( \gamma\right) =\mathbf{U}\left( \gamma\right) \cdot\mathbf{V} \left( \gamma\right)\tag{4.30}
and consider the difference
F(γ+h)F(γ)=U(γ+h)V(γ+h)U(γ)V(γ).(4.31)F\left( \gamma+h\right) -F\left( \gamma\right) =\mathbf{U}\left( \gamma+h\right) \cdot\mathbf{V}\left( \gamma+h\right) -\mathbf{U}\left( \gamma\right) \cdot\mathbf{V}\left( \gamma\right) .\tag{4.31}
On the right, subtract the combination U(γ)V(γ+h)\mathbf{U}\left( \gamma\right) \cdot\mathbf{V}\left( \gamma+h\right) from the first term and add it to the second, i.e.
F(γ+h)F(γ)=(U(γ+h)V(γ+h)U(γ)V(γ+h))+                                    (U(γ)V(γ+h)U(γ)V(γ)).          (4.32)\begin{aligned}F\left( \gamma+h\right) -F\left( \gamma\right) & =\left( \mathbf{U} \left( \gamma+h\right) \cdot\mathbf{V}\left( \gamma+h\right) -\mathbf{U}\left( \gamma\right) \cdot\mathbf{V}\left( \gamma+h\right) \rule{0pt}{12pt}\right) +\ \ \ \ \ \ \ \ \ \ \\& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( \mathbf{U} \left( \gamma\right) \cdot\mathbf{V}\left( \gamma+h\right) -\mathbf{U} \left( \gamma\right) \cdot\mathbf{V}\left( \gamma\right) \rule{0pt}{12pt} \right) .\ \ \ \ \ \ \ \ \ \ \left(4.32\right)\end{aligned}
By the distributive law,
F(γ+h)F(γ)=(U(γ+h)U(γ))V(γ+h)+                      U(γ)(V(γ+h)V(γ)).          (4.33)\begin{aligned}F\left( \gamma+h\right) -F\left( \gamma\right) & =\left( \mathbf{U} \left( \gamma+h\right) -\mathbf{U}\left( \gamma\right) \rule{0pt}{12pt} \right) \cdot\mathbf{V}\left( \gamma+h\right) +\ \ \ \ \ \ \ \ \ \ \\& \ \ \ \ \ \ \ \ \ \ \ \ \mathbf{U}\left( \gamma\right) \cdot\left( \mathbf{V}\left( \gamma+h\right) -\mathbf{V}\left( \gamma\right) \rule{0pt}{12pt}\right) .\ \ \ \ \ \ \ \ \ \ \left(4.33\right)\end{aligned}
Divide both sides by hh, i.e.
F(γ+h)F(γ)h=U(γ+h)U(γ)hV(γ+h)+U(γ)V(γ+h)V(γ)h.(4.34)\frac{F\left( \gamma+h\right) -F\left( \gamma\right) }{h}=\frac {\mathbf{U}\left( \gamma+h\right) -\mathbf{U}\left( \gamma\right) } {h}\cdot\mathbf{V}\left( \gamma+h\right) +\mathbf{U}\left( \gamma\right) \cdot\frac{\mathbf{V}\left( \gamma+h\right) -\mathbf{V}\left( \gamma\right) }{h}.\tag{4.34}
Finally, evaluate the limit as hh approaches zero. By definition, the left side approach
F(γ)=(U(γ)V(γ)).(4.35)F^{\prime}\left( \gamma\right) =\left( \mathbf{U}\left( \gamma\right) \cdot\mathbf{V}\left( \gamma\right) \rule{0pt}{12pt}\right) ^{\prime}.\tag{4.35}
Meanwhile, the two fractions on the right approach to U(γ)\mathbf{U}^{\prime }\left( \gamma\right) and V(γ)\mathbf{V}^{\prime}\left( \gamma\right) , and V(γ+h)\mathbf{V}\left( \gamma+h\right) approaches V(γ)\mathbf{V}\left( \gamma\right) . Thus, the right side converges to
U(γ)V(γ)+U(γ)V(γ)(4.36)\mathbf{U}^{\prime}\left( \gamma\right) \cdot\mathbf{V}\left( \gamma\right) +\mathbf{U}\left( \gamma\right) \cdot\mathbf{V}^{\prime }\left( \gamma\right)\tag{4.36}
leading to the identity
(U(γ)V(γ))=U(γ)V(γ)+U(γ)V(γ),(4.37)\left( \mathbf{U}\left( \gamma\right) \cdot\mathbf{V}\left( \gamma\right) \rule{0pt}{12pt}\right) ^{\prime}=\mathbf{U}^{\prime}\left( \gamma\right) \cdot\mathbf{V}\left( \gamma\right) +\mathbf{U}\left( \gamma\right) \cdot\mathbf{V}^{\prime}\left( \gamma\right) ,\tag{4.37}
as we set out to show.
As one demonstration of the dot product rule, let us show that if U(γ)\mathbf{U}\left( \gamma\right) has constant length then U(γ)\mathbf{U} ^{\prime}\left( \gamma\right) is orthogonal to U(γ)\mathbf{U}\left( \gamma\right) . In algebraic terms, the fact that U(γ)\mathbf{U}\left( \gamma\right) has constant length can be expressed with the help of the dot product by the equation
U(γ)U(γ)=c,(4.38)\mathbf{U}\left( \gamma\right) \cdot\mathbf{U}\left( \gamma\right) =c,\tag{4.38}
where cc is a constant. An application of the dot product rule yields
U(γ)U(γ)+U(γ)U(γ)=0.(4.39)\mathbf{U}^{\prime}\left( \gamma\right) \cdot\mathbf{U}\left( \gamma\right) +\mathbf{U}\left( \gamma\right) \cdot\mathbf{U}^{\prime }\left( \gamma\right) =0.\tag{4.39}
The two terms on the left are equal and therefore both equal zero. In other words, the dot product of U(γ)\mathbf{U}^{\prime}\left( \gamma\right) and U(γ)\mathbf{U}\left( \gamma\right) vanishes
U(γ)U(γ)=0.(4.40)\mathbf{U}^{\prime}\left( \gamma\right) \cdot\mathbf{U}\left( \gamma\right) =0.\tag{4.40}
In other words, U(γ)\mathbf{U}^{\prime}\left( \gamma\right) is indeed orthogonal to U(γ)\mathbf{U}\left( \gamma\right) .
Orthogonality between U(γ)\mathbf{U}\left( \gamma\right) and U(γ)\mathbf{U} ^{\prime}\left( \gamma\right) for a constant-length function U(γ)\mathbf{U}\left( \gamma\right) can also be derived from the previously observed fact that U(γ)\mathbf{U}^{\prime}\left( \gamma\right) is tangential to the curve traced out by U(γ)\mathbf{U}\left( \gamma\right) when the vectors U(γ)\mathbf{U}\left( \gamma\right) emanate from a fixed point OO. When the vectors represented by a constant-length function U(γ)\mathbf{U}\left( \gamma\right) are arranged in such a way, the tips of U(γ)\mathbf{U}\left( \gamma\right) trace out a circle, although not necessarily in a constant-speed fashion encountered in the previous Section. Since U(γ)\mathbf{U}^{\prime}\left( \gamma\right) is tangential to the circle, it must be orthogonal to U(γ)\mathbf{U}\left( \gamma\right) as the latter points in the radial direction.
For another interesting interpretation of orthogonality between a constant-length vector and its derivative, imagine a particle moving along a curved path with constant speed. Because the particle's trajectory is not straight, its velocity V(γ)=R(γ)\mathbf{V}\left( \gamma\right) =\mathbf{R}^{\prime}\left( \gamma\right) is not constant. However, the magnitude of the velocity, i.e. speed, can remain constant even along a curved trajectory.
(4.41)
As we have established previously, the velocity vector V(γ)\mathbf{V}\left( \gamma\right) is tangential to the trajectory. Its derivative V(γ)\mathbf{V}^{\prime}\left( \gamma\right) , which is orthogonal to V\mathbf{V}, is the acceleration A(γ)\mathbf{A}\left( \gamma\right) . Thus, the fact established in this Section shows that the acceleration of a particle moving with constant speed is orthogonal to the trajectory.
We will now turn our attention to the important concepts of the directional derivative and the gradient of a scalar field.
A field is a quantity defined at every point of a domain in a Euclidean space. We will consider fields of scalar, vector, and (eventually) variant quantities. An example of a scalar field is the temperature distribution in a room. The following figure shows a density plot of a two-dimensional scalar field, where the color of a given point corresponds to the value of the field. Meanwhile, the plotted contours, known as level sets, corresponds to a particular fixed value of the function defining the field.
(4.42)
An example of a vector field is the distribution of velocities in a fluid flow. A vector field is usually represented by plotting the vectors of the field at a strategic sampling of points.
(4.43)
The concept of the directional derivative can be applied to a field of any kind. As the name suggests, it measures the rate of change of the field in a particular direction.
Consider a scalar field UU defined in a Euclidean space. We will now construct its directional derivative at a point PP in the direction indicated by a unit vector L\mathbf{L}. Let ll be the ray that emanates from the point PP in the direction of L\mathbf{L}. The following figure shows these elements as well as the density plot for UU.
(4.44)
For a small positive number hh, find the point PP^{\ast} along ll whose Euclidean distance to PP equals hh. Denote the values of UU at the two points by U(P)U\left( P\right) and U(P)U\left( P^{\ast}\right) .
(4.45)
The difference U(P)U(P)U\left( P^{\ast}\right) -U\left( P\right) represents the change in UU from PP to PP^{\ast}, while the ratio
U(P)U(P)h(4.46)\frac{U\left( P^{\ast}\right) -U\left( P\right) }{h}\tag{4.46}
can be thought of as the average rate of change. As hh approaches 00, this quantity approaches the instantaneous rate of change of UU in the direction ll. This quantity is denoted by dU/dldU/dl and is known as the directional derivative of UU in the direction ll. In formal terms,
dUdl=limh0U(P)U(P)h.(4.47)\frac{dU}{dl}=\lim_{h\rightarrow0}\frac{U\left( P^{\ast}\right) -U\left( P\right) }{h}.\tag{4.47}
The same definition can be applied to a vector field textbf{UU}, i.e.
dUdl=limh0U(P)U(P)h.(4.48)\frac{d\mathbf{U}}{dl}=\lim_{h\rightarrow0}\frac{\mathbf{U}\left( P^{\ast }\right) -\mathbf{U}\left( P\right) }{h}.\tag{4.48}
Note, once again, that vectors are subject to each of the operations featured on the right side of this equation. Also note that, like most of the concepts introduced in this book so far, the directional derivative is defined in pure geometric terms without a reference to a coordinate system.
Rather than directly relying on the concept of a limit, the directional derivative can be defined in terms of the ordinary derivative. Identify with every point in the Euclidean space the position vector R\mathbf{R} emanating from an arbitrary origin OO so that the scalar field UU can be thought of as a function U(R)U\left( \mathbf{R}\right) of the position vector. Denote the position vector associated with the point PP by R0\mathbf{R}_{0}. Then the expression R0+sL\mathbf{R}_{0}+s\mathbf{L} captures the points along the ray ll while U(R0+sL)U\left( \mathbf{R}_{0}+s\mathbf{L}\right) captures the corresponding values of UU. Since L\mathbf{L} is a unit vector, the parameter ss represents the distance to the point PP. Thus, the directional derivative dU/dldU/dl can be defined as the ordinary derivative of U(R0+sL)U\left( \mathbf{R}_{0}+s\mathbf{L}\right) with respect to ss, i.e.
dUdl=ddsU(R0+sL).(4.49)\frac{dU}{dl}=\frac{d}{ds}U\left( \mathbf{R}_{0}+s\mathbf{L}\right) .\tag{4.49}
Naturally, the same idea applies to vector fields. For a vector field textbf{UU}, consider the vector-valued function textbf{UU}(R0+sL)\left( \mathbf{R}_{0}+s\mathbf{L}\right) and define ddtextbf{UU}/dl/dl as the derivative with respect to ss of textbf{UU}(R0+sL)\left( \mathbf{R} _{0}+s\mathbf{L}\right) .
Let us now consider several examples involving the directional derivative. In addition to serving as concrete illustrations of the concept, these examples will offer two further benefits. First, they will reinforce the pure geometric nature of our narrative. Second, they will yield results essential for a number of future applications, including the introduction of the covariant basis in Chapter 9 and of the Christoffel symbol in Chapter 12.

4.7.1Directional derivative example 11

For the first example, let U(P)U\left( P\right) be the Euclidean distance dd between PP and a fixed point OO, and determine dU/dldU/dl for ll that points directly away from OO.
(4.50)
The above figure illustrates U(P)U\left( P\right) and shows the points PP^{\ast} featured in the definition
dUdl=limh0U(P)U(P)h(4.47)\frac{dU}{dl}=\lim_{h\rightarrow0}\frac{U\left( P^{\ast}\right) -U\left( P\right) }{h} \tag{4.47}
of the direction derivative.
The distance between OO and PP^{\ast} is d+hd+h, i.e.
U(P)=d+h.(4.51)U\left( P^{\ast}\right) =d+h.\tag{4.51}
Consequently,
U(P)U(P)h=(d+h)dh=hh=1 for all h.(4.52)\frac{U\left( P^{\ast}\right) -U\left( P\right) }{h}=\frac{\left( d+h\right) -d}{h}=\frac{h}{h}=1\text{ for all }h.\tag{4.52}
Therefore,
limh0U(P)U(P)h=1.(4.53)\lim_{h\rightarrow0}\frac{U\left( P^{\ast}\right) -U\left( P\right) } {h}=1.\tag{4.53}
In words, the derivative dU/dldU/dl of UU in the direction away from OO equals 11 at all points, i.e.
dUdl=1.(4.54)\frac{dU}{dl}=1.\tag{4.54}

4.7.2Directional derivative example 22

For a second example, consider the same scalar field UU as in the previous example, but let the ray ll point in the counterclockwise orthogonal direction to the segment OPOP. The construction necessary for evaluating dU/dldU/dl is shown in the following figure.
(4.55)
By the Pythagorean theorem, the distance between OO and PP^{\ast} is d2+h2\sqrt{d^{2}+h^{2}}, i.e.
U(P)=d2+h2.(4.56)U\left( P^{\ast}\right) =\sqrt{d^{2}+h^{2}}.\tag{4.56}
Thus, dU/dldU/dl is given by
dUdl=limh0d2+h2dh.(4.57)\frac{dU}{dl}=\lim_{h\rightarrow0}\frac{\sqrt{d^{2}+h^{2}}-d}{h}.\tag{4.57}
It is a matter of ordinary Calculus to show that the limit vanishes and, therefore,
dUdl=0.(4.58)\frac{dU}{dl}=0.\tag{4.58}

4.7.3Directional derivative example 33

Let us now consider an example involving a vector field. Choose an arbitrary fixed point OO and let textbf{UU}(P)\left( P\right) be the vector that points in the counterclockwise direction orthogonal to the segment OPOP and has the length that equals the distance to the point OO. Calculate ddtextbf{UU}/dl/dl for ll pointing directly away from OO.
(4.59)
The following figure shows the values of the field textbf{UU} at two nearby points PP and PP^{\ast} along the ray ll.
(4.60)
The vectors textbf{UU}(P)\left( P\right) and textbf{UU}(P)\left( P^{\ast }\right) point in the same direction and, therefore, the difference U(P)U(P)\mathbf{U}\left( P^{\ast}\right) -\mathbf{U}\left( P\right) also points in that direction. Since the lengths of textbf{UU}(P)\left( P\right) and textbf{UU}(P)\left( P^{\ast}\right) are dd and d+hd+h, the length of U(P)U(P)\mathbf{U}\left( P^{\ast}\right) -\mathbf{U}\left( P\right) is hh. Consequently, the length of the ratio
U(P)U(P)h(4.61)\frac{\mathbf{U}\left( P^{\ast}\right) -\mathbf{U}\left( P\right) }{h}\tag{4.61}
is 11 for all hh. We can, therefore, conclude that ddtextbf{UU}/dl/dl is a unit vector pointing in the counterclockwise direction orthogonal to OPOP as illustrated in the following figure.
  (4.62)

4.7.4Directional derivative example 44

Finally, let us calculate the directional derivative of the same vector field textbf{UU} as in the previous example along the ray that points in the counterclockwise direction orthogonal to OPOP. The following figure shows two nearby points PP and PP^{\ast} along the ray ll and the corresponding vectors textbf{UU}(P)\left( P\right) and textbf{UU}(P)\left( P^{\ast }\right) .
(4.63)
This time, the distance between OO and PP^{\ast} is d2+h2\sqrt{d^{2}+h^{2}} and, therefore, the length of textbf{UU}(P)\left( P^{\ast}\right) is d2+h2\sqrt{d^{2}+h^{2}}. Shift the tail of textbf{UU}(P)\left( P^{\ast}\right) to the point PP and construct the difference U(P)U(P)\mathbf{U}\left( P^{\ast }\right) -\mathbf{U}\left( P\right) by connecting the tips of textbf{UU}(P)\left( P\right) and textbf{UU}(P)\left( P^{\ast}\right) .
(4.64)
Observe that the triangle OPPOPP^{\ast} is congruent to the triangle with the vertices at PP and the tips of textbf{UU}(P)\left( P\right) and textbf{UU}(P)\left( P^{\ast}\right) by the two sides and the included angle criterion. Consequently, the vector U(P)U(P)\mathbf{U}\left( P^{\ast}\right) -\mathbf{U}\left( P\right) is orthogonal to textbf{UU }(P)\left( P\right) and has length hh. Therefore, the vector
U(P)U(P)h(4.65)\frac{\mathbf{U}\left( P^{\ast}\right) -\mathbf{U}\left( P\right) }{h}\tag{4.65}
is orthogonal to textbf{UU}(P)\left( P\right) and has length 11. We thus conclude that ddtextbf{UU}/dl/dl is a unit vector that points towards OO, as illustrated in the following figure.
  (4.66)
We will now show that for any smooth scalar field UU, the directional derivative dU/dldU/dl at a point PP in the direction of the unit vector L\mathbf{L} is given by the equation
dUdl=GL,(4.67)\frac{dU}{dl}=\mathbf{G}\cdot\mathbf{L,}\tag{4.67}
where the vector G\mathbf{G} depends on the point PP but not the direction L\mathbf{L}. In the next Section, we will identify the vector G\mathbf{G} with the concept of the gradient.
Our derivation will rely on the fundamental idea underlying Calculus that in a small neighborhood of x0x_{0}, an ordinary function U(x)U\left( x\right) is given by
U(x)=U(x0)+k(xx0)+o(h),(4.68)U\left( x\right) =U\left( x_{0}\right) +k\left( x-x_{0}\right) +o\left( h\right) ,\tag{4.68}
where kk is the derivative of U(x)U\left( x\right) at x0x_{0}, hh equals xx0x-x_{0}, and o(h)o\left( h\right) is a quantity that approaches 00 faster than hh. In other words, over a "sufficiently small" interval, a smooth function U(x)U\left( x\right) is "essentially linear".
An analogous statement can be made for a scalar field in a Euclidean space. Namely, over a "sufficiently small" neighborhood, a scalar field UU is "essentially linear". In order to express this insight analytically, once again treat UU as a function of the position vector R\mathbf{R} emanating from an arbitrary point OO and denote the position vector associated with the point PP by R0\mathbf{R}_{0}. Then, in a small neighborhood of PP, U(R)U\left( \mathbf{R}\right) essentially equals the sum of its value at R0\mathbf{R} _{0} and a linear function of the difference RR0\mathbf{R} -\mathbf{R}_{0}. In other words, U(R)U\left( \mathbf{R}\right) is captured by the equation
U(R)=U(R0)+linear function of (RR0)+o(h),(4.69)U\left( \mathbf{R}\right) =U\left( \mathbf{R}_{0}\right) +\text{linear function of }\left( \mathbf{R}-\mathbf{R}_{0}\right) +o\left( h\right) ,\tag{4.69}
where hh is the length of RR0\mathbf{R}-\mathbf{R}_{0} and o(h)o\left( h\right) is once again a quantity that approaches 00 faster than hh.
As we demonstrated in Exercise 2.5, a linear function of a vector argument can be uniquely expressed by a dot product with a fixed vector G\mathbf{G}. Thus, U(R)U\left( \mathbf{R}\right) is given by
U(R)=U(R0)+G(RR0)+o(h).(4.70)U\left( \mathbf{R}\right) =U\left( \mathbf{R}_{0}\right) +\mathbf{G} \cdot\left( \mathbf{R}-\mathbf{R}_{0}\right) +o\left( h\right) .\tag{4.70}
If L\mathbf{L} is the unit vector pointing in the same direction as RR0\mathbf{R}-\mathbf{R}_{0}, i.e.
RR0=hL,(4.71)\mathbf{R}-\mathbf{R}_{0}=h\mathbf{L,}\tag{4.71}
then the previous identity can be rewritten as
U(R0+hL)=U(R0)+hGL+o(h).(4.72)U\left( \mathbf{R}_{0}+h\mathbf{L}\right) =U\left( \mathbf{R}_{0}\right) +h\mathbf{G}\cdot\mathbf{L}+o\left( h\right) .\tag{4.72}
Dividing both sides by hh, we find
U(R0+hL)U(R0)h=GL+o(h)h.(4.73)\frac{U\left( \mathbf{R}_{0}+h\mathbf{L}\right) -U\left( \mathbf{R} _{0}\right) }{h}=\mathbf{G}\cdot\mathbf{L}+\frac{o\left( h\right) }{h}.\tag{4.73}
As hh approaches 00, the left side approaches dU/dldU/dl while the right side approaches GL\mathbf{G}\cdot\mathbf{L}. Therefore, in the limit, we have
dUdl=GL,(4.67)\frac{dU}{dl}=\mathbf{G}\cdot\mathbf{L,} \tag{4.67}
as we set out to prove.
Comparing the equation
U(R0+hL)=U(R0)+hGL+o(h)(4.74)U\left( \mathbf{R}_{0}+h\mathbf{L}\right) =U\left( \mathbf{R}_{0}\right) +h\mathbf{G}\cdot\mathbf{L}+o\left( h\right)\tag{4.74}
with its "ordinary" counterpart
U(x0+h)=U(x0)+U(x0)h+o(h),(4.75)U\left( x_{0}+h\right) =U\left( x_{0}\right) +U^{\prime}\left( x_{0}\right) h+o\left( h\right) ,\tag{4.75}
we note that the vector G\mathbf{G} plays a role analogous to that of U(x0)U^{\prime}\left( x_{0}\right) . Thus, in a sense, we can think of G\mathbf{G} as the "vector derivative" of the function U(R)U\left( \mathbf{R}\right) . This interpretation will be further cemented by the introduction of the concept of the gradient, which is our next topic.

4.9.1A geometric definition

For scalar fields, the directional derivative leads to the crucial concept of the gradient of a scalar field UU. Let ll be the direction of the greatest increase in a scalar field UU at a point PP. Then, by definition, the gradient U\mathbf{\nabla}U, also denoted by gradU\operatorname{grad}U, is a vector of length dU/dldU/dl that points in the direction ll.
(4.76)
The concept of the gradient applies only to scalar fields since, as we discussed above, vectors cannot be compared in the same sense as numbers, i.e. for two vectors, there is no rule for determining which one is "greater".
Yet again, note the geometric nature of the newly introduced concept. For a given scalar field UU, the gradient can be evaluated, at least conceptually, by pure geometric means without a reference to a coordinate system. Thus, our approach differs from that found in most textbooks where the gradient is defined as a collection of partial derivatives. Later in this Chapter, we will begin the task of reconciling the two approaches.
In the previous Section, we showed that the directional derivative of a scalar field UU at a point PP in the direction of the unit vector L\mathbf{L} is given by the dot product
dUdl=GL,(4.67)\frac{dU}{dl}=\mathbf{G}\cdot\mathbf{L,} \tag{4.67}
where the vector G\mathbf{G} is independent of L\mathbf{L}. You may not be surprised to find out that G\mathbf{G} equals the gradient U\mathbf{\nabla} U. Since L\mathbf{L} is unit length, the definition of the dot product tells us that
dUdl=lenG cosγ,(4.77)\frac{dU}{dl}=\operatorname{len}\mathbf{G}\ \cos\gamma,\tag{4.77}
where γ\gamma is the angle between G\mathbf{G} and L\mathbf{L}. Since the greatest value of cosγ\cos\gamma is 11 and occurs when γ=0\gamma=0, the greatest possible value of dU/dldU/dl is
lenG,(4.78)\operatorname{len}\mathbf{G,}\tag{4.78}
and also occurs when γ=0\gamma=0, i.e. when the unit vector L\mathbf{L} points in the direction of G\mathbf{G}. Thus, the vector G\mathbf{G} indeed represents the direction of the greatest increase in UU and, furthermore, has the magnitude that equals the rate of the greatest increase. In other words, G\mathbf{G} is the gradient of UU, as we set out to show. Having established this important connection, we can rewrite the directional derivative formula
dUdl=GL,(4.67)\frac{dU}{dl}=\mathbf{G}\cdot\mathbf{L,} \tag{4.67}
in the form
dUdl=UL.(4.79)\frac{dU}{dl}=\mathbf{\nabla}U\cdot\mathbf{L}.\tag{4.79}
One of the insights offered by this formula is the fact that knowing the value of the gradient at a given point is sufficient for determining the directional derivative dU/dldU/dl in any direction ll. Furthermore, the directional derivative of a scalar field is 00 in any direction orthogonal to the gradient. Conversely, if the directional derivative in a particular direction ll is 00, then the gradient is orthogonal to ll, provided that the gradient is not zero. In particular, the gradient is orthogonal to the level sets of UU.
For the first example, again consider the function U(P)U\left( P\right) defined as the distance between the point PP and an arbitrary fixed point OO. It is intuitively clear that the gradient of U(P)U\left( P\right) at a point PP is a unit vector that points directly away from the point OO.
(4.80)
It is left as an exercise to demonstrate this fact analytically. Note that the gradient field U\mathbf{\nabla}U is not defined at OO, where it experiences a non-removable discontinuity.
For a second example, choose a fixed point OO along with a ray ll emanating from OO, and let U(P)U\left( P\right) be the angle between the segment OPOP and the ray ll, subject to the condition that the angle varies between 00 and 2π2\pi and is measured in the counterclockwise direction.
(4.81)
It is intuitively clear that the gradient points in the counterclockwise direction orthogonal to the segment OPOP and that its magnitude is inversely proportional to the distance dd between OO and PP. To calculate the precise magnitude of U\mathbf{\nabla}U, note that a step of hh in the counterclockwise orthogonal direction from PP results in the change in UU that equals arctan(h/d)\arctan\left( h/d\right) .
(4.82)
Thus, rate of change is given by the limit
limh0arctan(h/d)h.(4.83)\lim_{h\rightarrow0}\frac{\arctan\left( h/d\right) }{h}.\tag{4.83}
It is a matter of ordinary Calculus to show that this limit equals 1/d1/d, i.e.
lenU=1d.(4.84)\operatorname{len}\mathbf{\nabla}U=\frac{1}{d}.\tag{4.84}
The resulting gradient field is illustrated in the following figure.
(4.85)
Once again, U\mathbf{\nabla}U is discontinuous at OO.
In all likelihood, in your first encounter with the gradient, it was defined as the collection of partial derivatives
(Ux,Uy,Uz)(4.86)\left( \frac{\partial U}{\partial x},\frac{\partial U}{\partial y} ,\frac{\partial U}{\partial z}\right)\tag{4.86}
with respect to Cartesian coordinates x,y,zx,y,z. From this definition, it is then demonstrated that the elements of U\mathbf{\nabla}U represent the components of a vector that points in the direction of the greatest increase of UU, while the magnitude of that vector equals the corresponding greatest rate of increase. In other words, what we have adopted as the definition of the gradient appears as a consequence in the conventional approach.
Given that we now have two alternative definitions of the gradient, we must find a way to reconcile them. Furthermore, we ought to generalize the coordinate space expression for the gradient to general non-Cartesian coordinates. Looking ahead, the former task will be accomplished in Chapter 6 on coordinate systems while the latter will be accomplished in Chapter 9 in which some of the most fundamental coordinate-dependent objects will be introduced.
Exercise 4.1Show that for a circle of radius rr given by the vector equation of the curve R(γ)\mathbf{R}\left( \gamma\right) , where γ\gamma is the central angle, R(γ)\mathbf{R}^{\prime}\left( \gamma\right) is a vector of length rr.
Exercise 4.2Show that the derivative of vector-valued functions satisfies the sum rule
(U+V)=U+V.(4.26)\left( \mathbf{U}+\mathbf{V}\right) ^{\prime}=\mathbf{U}^{\prime} +\mathbf{V}^{\prime}. \tag{4.26}
Exercise 4.3Show that for a constant number cc, the derivative of vector-valued functions satisfies the rule
(cU)=cU.(4.27)\left( c\mathbf{U}\right) ^{\prime}=c\mathbf{U}^{\prime}. \tag{4.27}
Exercise 4.4Show that for a scalar function c(γ)c\left( \gamma\right) , the derivative of vector-valued functions satisfies the rule
(cU)=cU+cU.(4.28)\left( c\mathbf{U}\right) ^{\prime}=c^{\prime}\mathbf{U}+c\mathbf{U} ^{\prime}. \tag{4.28}
Exercise 4.5Show that for a scalar function ξ(γ)\xi\left( \gamma\right) , the derivative of the composite vector-valued function U(ξ(γ))\mathbf{U}\left( \xi\left( \gamma\right) \right) satisfies the chain rule
ddγU(ξ(γ))=U(ξ(γ))ξ(γ).(4.28)\frac{d}{d\gamma}\mathbf{U}\left( \xi\left( \gamma\right) \right) =\mathbf{U}^{\prime}\left( \xi\left( \gamma\right) \right) \xi^{\prime }\left( \gamma\right) . \tag{4.28}
Exercise 4.6Show that the derivative of vector-valued functions applied to the cross product satisfies the product rule
(U×V)=U×V+U×V.(4.87)\left( \mathbf{U}\times\mathbf{V}\right) ^{\prime}=\mathbf{U}^{\prime} \times\mathbf{V}+\mathbf{U}\times\mathbf{V}^{\prime}.\tag{4.87}
Exercise 4.7For a vector-valued function R(γ)\mathbf{R}\left( \gamma\right) that traces out the unit circle, describe R(γ)\mathbf{R}^{\prime\prime}\left( \gamma\right) and R(γ)\mathbf{R}^{\prime\prime\prime}\left( \gamma\right) in geometric terms.
Exercise 4.8In Section 4.5, we showed that the acceleration of a particle moving with constant speed is orthogonal to its trajectory. Show that the converse is also true: if the acceleration of a particle is orthogonal to its trajectory, then its speed is constant.
Exercise 4.9Consider a particle moving with constant speed. Show that
len2A(t)=V(t)V(t),(4.88)\operatorname{len}^{2}\mathbf{A}\left( t\right) =-\mathbf{V}^{\prime\prime }\left( t\right) \cdot\mathbf{V}\left( t\right) ,\tag{4.88}
where A(t)=V(t)\mathbf{A}\left( t\right) =\mathbf{V}^{\prime}\left( t\right) is the acceleration vector.
Exercise 4.10Show that the trajectory of a particle moving with constant speed and acceleration of constant magnitude is a circle.
Exercise 4.11Given two fixed points AA and BB in a three-dimensional space, let U(P)U\left( P\right) be the area of the triangle ABPABP. Evaluate dU/dldU/dl for L\mathbf{L} pointing in the direction parallel to ABAB.
Exercise 4.12For the same function U(P)U\left( P\right) , evaluate dU/dldU/dl for L\mathbf{L} pointing in the direction orthogonal to and away from ABAB within the plane of the triangle ABPABP.
Exercise 4.13For the same function U(P)U\left( P\right) , describe the gradient U\mathbf{\nabla}U in geometric terms. Show that if the vectors P\mathbf{P}, A\mathbf{A}, and B\mathbf{B} correspond to the points PP, AA, and BB, then U\mathbf{\nabla}U points in the direction of the vector
P+(PA)(AB)(AB)(AB)B+(PB)(BA)(BA)(BA)A(4.89)\mathbf{P}+\frac{\left( \mathbf{P}-\mathbf{A}\right) \cdot\left( \mathbf{A}-\mathbf{B}\right) }{\left( \mathbf{A}-\mathbf{B}\right) \cdot\left( \mathbf{A}-\mathbf{B}\right) }\mathbf{B}+\frac{\left( \mathbf{P}-\mathbf{B}\right) \cdot\left( \mathbf{B}-\mathbf{A}\right) }{\left( \mathbf{B}-\mathbf{A}\right) \cdot\left( \mathbf{B}-\mathbf{A} \right) }\mathbf{A}\tag{4.89}
and has the same magnitude as the vector (BA)/2\left( \mathbf{B}-\mathbf{A} \right) /2.
Exercise 4.14Given two fixed points AA and BB, let U(P)=AP+BPU\left( P\right) =AP+BP. Describe the direction and the magnitude of U\mathbf{\nabla}U, illustrated in the following figure, in geometric terms.
(4.90)
Exercise 4.15Confirm that dU/dl=ULdU/dl=\mathbf{\nabla}U\cdot\mathbf{L} for the four examples in Section 4.7.
Exercise 4.16Given a point AA and a curve Γ\Gamma, show that if the point BB is the closest to AA among all points on Γ\Gamma, then the segment ABAB is orthogonal to Γ\Gamma.
(4.91)
Exercise 4.17Given two non-intersecting curves Γ1\Gamma_{1} and Γ2\Gamma_{2}, show that if the segment ABAB represents the shortest distance between the two curves, then ABAB is orthogonal to each curve.
(4.92)
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