# 13 - Contents CHAPTER 9 9.1 9.2 9.3 9.4 Polar Coordinates...

This preview shows pages 1–3. Sign up to view the full content.

Contents CHAPTER 9 9.1 9.2 9.3 9.4 CHAPTER 10 10.1 10.2 10.3 10.4 10.5 CHAPTER 11 11.1 11.2 11.3 11.4 11.5 CHAPTER 12 12.1 12.2 12.3 12.4 CHAPTER 13 13.1 13.2 13.3 13.4 13.5 13.6 13.7 Polar Coordinates and Complex Numbers Polar Coordinates 348 Polar Equations and Graphs 351 Slope, Length, and Area for Polar Curves 356 Complex Numbers 360 Infinite Series The Geometric Series Convergence Tests: Positive Series Convergence Tests: All Series The Taylor Series for ex, sin x, and cos x Power Series Vectors and Matrices Vectors and Dot Products Planes and Projections Cross Products and Determinants Matrices and Linear Equations Linear Algebra in Three Dimensions Motion along a Curve The Position Vector 446 Plane Motion: Projectiles and Cycloids 453 Tangent Vector and Normal Vector 459 Polar Coordinates and Planetary Motion 464 Partial Derivatives Surfaces and Level Curves 472 Partial Derivatives 475 Tangent Planes and Linear Approximations 480 Directional Derivatives and Gradients 490 The Chain Rule 497 Maxima, Minima, and Saddle Points 504 Constraints and Lagrange Multipliers 514

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
CHAPTER 13 Partial Derivatives This chapter is at the center of multidimensional calculus. Other chapters and other topics may be optional; this chapter and these topics are required. We are back to the basic idea of calculus-the derivative. There is a functionf, the variables move a little bit, and f moves. The question is how much f moves and how fast. Chapters . 1-4 answered this question for f(x), a function of one variable. Now we have f(x, y) or f(x, y, z)-with two or three or more variables that move independently. As x and y change, f changes. The fundamental problem of differential calculus is to connect Ax and Ay to Af. Calculus solves that problem in the limit. It connects dx and dy to df. In using this language I am building on the work already done. You know that dfldx is the limit of AflAx. Calculus computes the rate of change-which is the slope of the tangent line. The goal is to extend those ideas to fix, y) = x2 - y2 or = Jm or f(x, y, z) = 2x + 3y + 42. These functions have graphs, they have derivatives, and they must have tangents. The heart of this chapter is summarized in six lines. The subject is diflerential calculus-small changes in a short time. Still to come is integral calculus-adding up those small changes. We give the words and symbols for f(x, y), matched with the words and symbols for f(x). Please use this summary as a guide, to know where calculus is going. Curve y = f(x) vs. Surface z = df becomes two partial derivatives - af and - af d~ ax ay - becomes four second derivatives - -- - d2{ a2f ax2' axayY ayai ay2 Af % AX becomes the linear approximation % 9 AX + a f ~ ~ ax ay tangent line becomes the tangent plane z - z, = af(x - x,) + af(y - yo) ax ay dy - dy dx dz az'ax a~ dy ---- becomes the chain rule - = - - +-- dt d~ dt dt a~ dt a~ dt - df = 0 becomes two maximum-minimum equations - af = 0 and af = 0.
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 06/13/2009 for the course TAM 455 taught by Professor Petrina during the Fall '08 term at Cornell.

### Page1 / 51

13 - Contents CHAPTER 9 9.1 9.2 9.3 9.4 Polar Coordinates...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online