Calculus: Analytic Geometry and Calculus, with Vectors

546 Polar, cylindrical, and spherical coordinates which may help us determine x(t) and y(t). Using the fact that the distance fr ...

10.3 Areas and integrals involving polar coordinates 547 is called the witch of Agnesi because Maria Gaetena Agnesi (1718-1799) ...

548 Polar, cylindrical, and spherical coordinates in which the subscripts and star superscripts do not appear. In applica- tions ...

10.3 Areas and integrals involving polar coordinates 549 in the negative direction. We never have negative areas, but we can sub ...

550 Polar, cylindrical, and spherical coordinates 9 It is much easier to learn to play a violin than to acquire competence to gi ...

10.3 Areas and integrals involving polar coordinates 551 in such a way that 0 is a function of t having two derivatives, then th ...

552 Polar, cylindrical, and spherical coordinates Q1PQ2 is a point of S. No interior point Q of the shaded sector of Figure 10.3 ...

11 Partial derivatives 11.1 Elementary partial derivatives complicated situations to later sections, we confine attention in thi ...

554 Partial derivatives at the "space-time place" having "space-time coordinates" x and t. Thus the temperature u is a function ...

11.1 Elementary partial derivatives 555 books and on blackboards and in notebooks and on scratch pads. Wher- ever we find aw/aq, ...

556 Partial derivatives which serve a dual purpose: they provide abbreviations for the expressions on the left sides, and they p ...

11.1 Elementary partial derivatives 557 Definition 11.16 14 function f of n variablesx1, x2, , x is said to have the limit L as ...

558 Partial derivatives Supposing for the moment that y and y + Ay remain fixed like good numbers usually do, we define a functi ...

11.1 Elementary partial derivatives 559 us to make any change we please in order of differentiation when u is differentiated mor ...

(^560) Partial derivatives satisfies the Laplace equation over the entire plane but is, nevertheless, not harmonic over regions ...

11.1 Elementary partial derivatives 561 is called the divergence of V. Finally, the vector function V X V defined by i j k (4) V ...

562 12 The function u for which u(0,0) = 0 and 2xy (1) u(x,Y) =(x2 +y2)2 can be considered more than once. Show that Partial der ...

11.2 Increments, chain rule, and gradients 563 ticular direction of a vector. In any case, it should be recognized that our func ...

564 Partial derivatives Looking forward to derivation of a formula (the chain formula) for w'(t) we write Thus where w(t + At) = ...

11.2 Increments, chain rule, and gradients 565 linked to t by the intermediate variables x, y, z which are functions of t. We ca ...