Gradients of Vector-Valued Functions

Functions of Three or More Variables

5.3 Gradients of Vector-Valued Functions

Vector-Valued Functions

Partial Derivative of Vector-Valued Functions

Jacobian

Change-of-variable

We can find this scaling factor by finding a mapping that transforms the unit square into the other square. In linear algebra terms, we effectively perform a variable transformation from (b1; b2) to (c1; c2). In our case, the mapping is linear and the absolute value of the determinant of this mapping gives us exactly the scaling factor we are looking for.

We will describe two approaches to identify this mapping. First, we exploit that the mapping is linear so that we can use the tools from Chapter 2

to identify this mapping. Second, we will find the mapping using partial derivatives using the tools we have been discussing in this chapter.

Scaling factor

The Jacobian represents the coordinate transformation we are looking for. It is exact if the coordinate transformation is linear (as in our case), and (5.66) recovers exactly the basis change matrix in (5.62). If the

coordinate transformation is nonlinear, the Jacobian approximates this

nonlinear transformation locally with a linear one. The absolute value of the Jacobian determinant |det(J)| is the factor by which areas or volumes are scaled when coordinates are transformed. Our case yields |det(J)| = 3.

Jacobian determinant gives the magnification/scaling factor whenwe transform an area or volume.

The Jacobian determinant and variable transformations will become

relevant in Section 6.7 when we transform random variables and probability distributions. These transformations are extremely relevant in machine

learning in the context of training deep neural networks using the reparametrization trick, also called infinite perturbation analysis.

Dimensionality of (partial) derivatives

Example 5.9

Example 5.11 (linear regression)

Change of Variables in Multiple Integrals

In one-dimensional calculus we often use a change of variable (a substitution) to simplify an integral. By reversing the roles of x and u, we can write the

Substitution Rule as

where x = g(u) and a = g(c), b = g(d). Another way of writing Formula 1 is as follows:

Change of Variables in Multiple Integrals

More generally, we consider a change of variables that is given by a transformation T from the uv-plane to the

xy-plane:

where x and y are related to u and v by the equations

or, as we sometimes write,

Change of Variables in Multiple Integrals

We usually assume that T is a C¹ transformation, which means that g and h have continuous first-order partial derivatives.

A transformation T is really just a function whose domain and range are both subsets of . If , then the point (x₁, y₁) is called the image of the point (u₁, v₁).

If no two points have the same image, T is called one-to-one.

Change of Variables in Multiple Integrals

Figure 1 shows the effect of a transformation T on a region S in the uv-plane. T transforms S into a region R in the

xy-plane called the image of S, consisting of the images of all points in S.

Change of Variables in Multiple Integrals

If T is a one-to-one transformation, then it has an inverse transformation T^–1 from the xy-plane to the uv-plane and it may be possible to solve Equations 3 for u and v in terms of x and y:

Now let’s see how a change of variables affects a double integral.

Change of Variables in Multiple Integrals

We start with a small rectangle S in the uv-plane whose lower left corner is the point (u₀, v₀) and whose dimensions are and . (See Figure 3.)

Figure 3

Change of Variables in Multiple Integrals

The image of S is a region R in the xy-plane, one of whose boundary points is .

The vector

is the position vector of the image of the point (u, v). The Jacobian of the transformation is given a special notation.

Change of Variables in Multiple Integrals

With this notation we can use Equation 6 to give an approximation to the area of R:

where the Jacobian is evaluated at (u₀, v₀).

Change of Variables in Multiple Integrals

Next we divide a region S in the uv-plane into rectangles S_ij and call their images in the xy-plane R_ij. (See Figure 6.)

Figure 6

Change of Variables in Multiple Integrals

Applying the approximation to each R_ij, we approximate the double integral of f over R as follows:

where the Jacobian is evaluated at (u_i, v_j). Notice that this double sum is a Riemann sum for the integral

Change of Variables in Multiple Integrals

Theorem 9 says that we change from an integral in x and y to an integral in u and v by expressing x and y in terms of u and v writing

Example 3

Evaluate the integral , where R is the trapezoidal region with vertices (1, 0), (2, 0), (0, –2), and

(0, –1).

Solution:

Since it isn’t easy to integrate , we make a change of variables suggested by the form of this function:

These equations define a transformation T^-1 from the

xy-plane to the uv-plane. Theorem 9 talks about a transformation T from the uv- plane to the xy-plane.

Example 3 – Solution

It is obtained by solving Equations 10 for x and y:

The Jacobian of T is

cont’d

Example 3 – Solution

To find the region S in the uv-plane corresponding to R, we note that the sides of R lie on the lines

and, from either Equations 10 or Equations 11, the image lines in the uv-plane are

cont’d

Example 3 – Solution

Thus the region S is the trapezoidal region with vertices (1, 1), (2, 2), (–2, 2), and (–1, 1) shown in Figure 8.

cont’d

Figure 8

Dalam dokumen Single and Multivariable Calculus (Halaman 65-98)