Transformations and A Universal First Order Taylor Expansion

Transcription

1 Transformations and A Universal First Order Taylor Expansion MATH 1502 Calculus II Notes September 29, 2008 The first order Taylor approximation for f : R R at x = x 0 is given by P 1 (x) = f (x 0 )(x x 0 ) + f(x 0 ), and for f : R 2 R the first order Taylor approximation is given by P 1 (x) = Df(x 0 )(x x 0 ) + f(x 0 ). These should be familiar to you by now, and you should know what it means that they are affine approximations. You should also be able to identify the center of expansion and the affine center of approximation and you should know which spaces contain them. Finally, you should be able to identify the linear part. The Differential Incidentally, the linear part of a first order Taylor approximation has a special name, when considered as a linear function. It is called the differential and is denoted by df x0. Notice that the subscript tells you the center of expansion. Thus, df x0 (v) = Df(x 0 ) v. 1

2 Exercise 1. A right circular cone has height 10 meters and radius 5 meters. If the radius is increased by 1 cm, use a first order Taylor approximation to estimate how much the height should be decreased to keep the volume the same. Identify the proper function, specify the center of expansion, affine center and the linear part of the approximation. 2. Find the exact answer to the last question. 3. Rewrite the first order affine approximation formulas above using differentials. One of the things we are going to cover in these notes is a first order affine approximation formula for any function with domain in R n and target R m. Let s first talk a little bit about such functions. They may seem somewhat exotic at first sight, so we ll give them a fancy name and notation. If F : R n R m, we will call F a transformation. Just like the temperature function T(x, y, z), which gives the temperature at a location (x, y, z) in a room, is a good example of a function from R 3 to R 1, there are many good examples of transformations associated with flowing liquids. Let us assume that a liquid flows in a region in R 3. Think of a point in the domain as a starting point x for a particle that flows with the liquid. After some fixed unit of time, we take a snapshot of the liquid and the position of the particle is F(x). Just to repeat that a different way: The ending position F of a particle embedded in a flowing liquid which starts at x = (x, y, z) at time zero and flows for a specified unit of time is a transformation of R 3 into itself. We may also wish to consider such a flow at more than two times. If we say that P is the position of a particle which starts at x at time zero and ends up at P at time t, then we also get a transformation. In this case, P : R 4 R 3 because P = P(x, y, z, t). Similarly, we could talk about the velocity V of the fluid, which would also take values in R 3. Another beautiful collection of examples is provided by conformal maps. These are functions of two variables that move the plane around in such 2

3 a way that angles are preserved. Most of the flat representations of the Earth s surface (Mercator, azimuthal, cylindrical, stereographic) have this angle preserving property. Because of this, if you look at two rivers (or roads or any two curves) that intersect in a certain angle on one map, they will intersect in the same angle on the other map (and on the Earth). In this case, we get a transformation F : R 2 R 2. Let s be a little more precise and make sure we understand this because we are going to use it as a central example below. Let us call one map the original map M 1 and the other the image map M 2. Then really, M 1 R 2 and F : M 1 R 2 with range M 2. The transformation will look something like F(x, y) = (u(x, y), v(x, y)) where u(x, y) is the x-coordinate of the point (on the image map) corresponding to (x, y) on the original map and v(x, y) is the y-coordinate of the point (on the image map) corresponding to (x, y) on the original map. We will mostly be interested in transformations in which the domain dimension and target dimension are the same, F : R n R n, but there is nothing that limits most of what we say to this case. So we could ask the question: Given a conformal map, what is the first order affine approximation for it? Obviously, the zero order approximation is just the function value: F(x) F(x 0 ). But what is the next term? It should be some kind of linear function from R 2 to R 2 applied to the affine term x x 0 : F(x) F(x 0 ) + L(x x 0 ). We saw in the last set of exercises that all such linear functions L are given by matrix multiplication. So we are asking what is the matrix that tells us what F is doing to first order. The answer, as usual, is that the matrix is the derivative. So what is the derivative? In this case, it is DF = u x v x As usual, we evaluate this at the center of expansion. Thus, a universal formula for first order Taylor approximation at x 0 is given by u y v y. F(x) F(x 0 ) + DF(x 0 )(x x 0 ). 3

4 The good news about this formula is that it doesn t look much different from the one in from calculus one (when f : R 1 R 1 ) or the one for real valued functions of two variables. It will always be the case that the derivative is a matrix whose entries are the partial derivatives of the component functions of F. We do have to pay a little more attention to how those partial derivatives are arranged in the matrix and how the matrix DF(x 0 ) is multiplied by the vector x x 0. (Hint: This kind of matrix multiplication has already been discussed in the handout on power series in two variables; write the vectors as column vectors.) The best way to do this is probably by doing some exercises. Exercise 4. The function F(x, y) = (3x, 3y) is called a dilation. Describe the image of the plane R 2 under this function. 5. Find the derivative of the dilation in the last exercise. 6. The function F(x, y) = ( x, y) is called a reflection. Describe the image of the plane R 2 under this function. 7. Find the first order Taylor approximation of the reflection in the last problem. Use the origin as the center of expansion. 8. What is the formula for a function which reflects R 2 about the x-axis? 9. Describe what the function F(x, y) = ( 3x, 3y) does to the plane. 10. Find the first order Taylor approximation of this function at the origin. 11. Which of the functions in exercises 4-10 are linear? Your answer to exercise 10 should look like ( ) ( 3x 3 0 3y 0 3 ) ( x y Here is the general definition for a transformation F : R n R m : Definition 1 The full derivative of a transformation F is the matrix ( ) fj x i ij where f j is the j-th component function of F. ). 4

5 Exercise 12. A closed rectangular box 4 feet long, 2 feet wide, 3 feet high is covered by a coat of paint 1/64 of an inch thick. Estimate the amount of paint on the box. Identify the function of interest, the center of expansion, and the differential. 13. Let u(x, y) = x 2 y 2 and v(x, y) = 2xy. Describe what the transformation F = (u, v) does to the plane. 14. Find the first order Taylor approximation centered at (x 0, y 0 ) for the function given in the last exercise. 15. Consider the transformation F(x, y) = (e x cos(y), e x sin(y)). Describe what this transformation does to R Find the first order Taylor approximation centered at (x 0, y 0 ) of the transformation in the last problem. 5