Vector functions of a vector variable, directional derivatives

VECTOR FUNCTIONS OF A VECTOR VARIABLE, DIRECTIONAL DERIVATIVES

Vector functions of a vector variable. The system

can be abbreviated as

and viewed as a mapping of vectors of some domain D in a vector space V into vectors in a vector space W. It maps an n-tuple (x₁, x₂, ... ,x_n) in V into an m-tuple (y₁, y₂, ... ,y_m) in W. Thus f can be viewed as a vector function of a vector variable assigning vectors to vectors. The independent variable is a vector and the dependent variable is a vector.

Example. In Differential Geometry the usual way of defining a surface in space is through parametric equations of the form

x = f(u, v)

3) y = g(u, v)

z = h(u, v)

where f, g, h are continuous functions defined on a simply connected region R of the uv-plane. We view this as a mapping from region R of the uv-plane into xyz-space. Let us re-write 3) with a different choice of notation as

y₁ = f₁(x₁, x₂)

4) y₂ = f₂(x₁, x₂)

y₃ = f₃(x₁, x₂)

where instead of the variables u, v we use x₁, x₂ and instead of x, y, z we use y₁, y₂, y₃. We can then abbreviate 4) as

Directional derivative (of a vector function of a vector variable). In Vector Analysis we encounter the concept of the derivative of a vector function of a real variable. We will now define the concept of the directional derivative of a vector function of a vector variable. Let a vector function of a vector variable

be defined on a domain D of a vector space V. The directional derivative of the function f is the derivative of f at a specified point in the domain computed for a specified direction.

Let be a vector in V. Let be a nonzero vector in V. The directional derivative is defined as follows:

Def. Directional derivative. The directional derivative of the function f at the point in the direction is the vector

whenever the limit exists.

In evaluating the limit in 7) we regard f as a function of h along the line . Let us denote the function by F(h). Then

Thus

i.e. the directional derivative in the direction of is given by the derivative of F(h) = with respect to h evaluated at h = 0.

For an intuitive understanding of what the directional derivative is let us consider a surface S in space defined by the parametric equations

x = f(u, v)

10) y = g(u, v)

z = h(u, v)

where f, g, h are continuous functions defined on a simply connected region R of the uv-plane. See Fig. 1. Let us again change notation as we did before and write 10) as

y₁ = f₁(x₁, x₂)

11) y₂ = f₂(x₁, x₂)

y₃ = f₃(x₁, x₂)

where instead of the variables u, v we use x₁, x₂ and instead of x, y, z we use y₁, y₂, y₃. We can then abbreviate 4) as

and view it as a vector function of a vector variable. We thus view as the position vector of a point P in the uv-coordinate system and as the position vector of the image of point P in the xyz-coordinate system. In Fig.1 note that traces out, as h varies, a line through in the direction of and that , the image of the line, is a curve on surface S.

It can thus be seen that the directional derivative

is a vector that is tangent to the curve at the point

Problem. Compute the directional derivative of the function

in the direction of a vector .

Solution. The function has been given in the form

which is equivalent to the system

y₁ = f₁(x₁, x₂)

15) y₂ = f₂(x₁, x₂)

y₃ = f₃(x₁, x₂)

We compute the directional derivative in a three step procedure:

Step 1. Form the function F(h). In the general problem, the functions f₁(x₁, x₂, ...), f₂(x₁, x₂, ...), etc. are functions of the components x₁, x₂, ... of the vector . We form F(h) by replacing, in each function f_j(x₁, x₂, ...) and for each x_i, all occurrences of the component x_i with the expression (x_i + hu_i). So replacing x₁ by (x₁ + hu₁) and x₂ by (x₂ + hu₂) in 13) we obtain

Step 2. Compute dF(h)/dh. Computing the derivative of 17) with respect to h we get

Step 3. Evaluate dF(h)/dh at h = 0. Evaluating 18) at h = 0 we get

Vectors, bases and coordinate systems. In general, vectors in a vector space are referred, either implicitly or explicitly, to some basis. In a space of n dimensions a basis can consist of any n linearly independent vectors. One chooses any n linearly independent vectors that he pleases and then these n vectors act as kind of framework or oblique coordinate system to which all other vectors are referenced. Since in spaces of dimension greater that three the regular type of coordinate system (i.e. a rectangular Cartesian system) is not possible, this is the type of “coordinate system” that is employed. In the space of three dimensions this type of coordinate system would consist of any set of three linearly independent vectors (i.e. three vectors not all lying in the same plane) and would constitute an “oblique” coordinate system in which the coordinate axes were, in general, not perpendicular to each other and, moreover, with units in the direction of the vectors, that, instead of being unity as in a Cartesian system, are the length of the vectors. There is, however, in an n-dimensional space, one particular basis, called the natural basis, that is implied if no other basis is explicitly stated. It is the basis consisting of the elementary unit vectors

e₁ = (1,0, ..., 0)

e₂ = (0,1, ..., 0)

........................

e_n = (0,0, ..., 1) .

In three dimensional space these elementary unit vectors correspond to the three unit vectors in the direction of the x, y and z axes familiar from Vector Analysis. In representing vector functions of the type we have been considering, , functions are often represented in the form

where the are the m basis vectors. Usually these basis vectors will be the elementary unit vectors (corresponding to of three dimensional space). Also, the independent variable will expressed in terms of basis vectors i.e.

where the will generally be the elementary unit vectors.

Directional derivative in the direction of a basis vector.

Theorem. Given the function where

and

Then the derivative of at point in the direction of the basis vector e_k is equal to the partial derivative of with respect to the k-th component of i.e.

Proof.

Example. Let

Then

Class of a vector function. If a function is continuous over its domain D, we say that it belongs to class C ⁰ in D. If all its first order derivatives

exist and are continuous we say that it belongs to class C ¹ in D. A vector function f is of class C ^m in a domain D if all m-th order derivatives

exist and are continuous in D.

If a function is of order C ^m it is also of order C ⁰, C ¹, ... , C ^m-1.

Theorem 1. If a function f is of class C ^m then in an m-th order derivative the order of differentiation is immaterial e.g.

Formula for the directional derivative of a function of class C ¹.

Theorem 2. Let be a function of class C ¹. Then the derivative of in the direction is given by

where x₁, x₂, ..., x_n are the n components of and u₁, u₂, ..., u_n are the n components of .

Example. Let

Then

Higher order directional derivatives. Suppose a function has a derivative in a fixed direction at every point in its domain D. Then itself is a function of in D and we can consider its derivative in a direction at . , if it exists, is called a second order directional derivative of f at and is denoted by Higher order derivatives are defined in the same way. For example, is a third order directional derivative at and is denoted by .

Components of second order derivatives in the direction of the basis vectors. Let (e₁, e₂, ... , e_n) be the basis in V. Then

Then

Thus we see that the components of the second order derivatives in the direction of the basis vectors are the second order partial derivatives of the function components f₁, f₂, ... , f_n.

Formulas for directional derivatives of higher orders. Suppose is of class C ² in its domain D. Then, from Theorem 2,