Calculus: Analytic Geometry and Calculus, with Vectors

566 Partial derivatives as valuable as the one that the rightmember of (11.242) is the scalar product of two vectors. Thus (11.2 ...

11.2 Increments, chain rule, and gradients 567 (isothermal surfaces or equipotential surfaces, for example) upon which u has a p ...

568 Partial derivatives can we estimate the error resulting from using f(xo,yo,zo) instead of f(x,y,z)? 11ns.: Put x=xo, Y=Yo, z ...

11.2 Increments, chain rule, and gradients 569 summation signs, we can easily extend all of these calculations to cover more com ...

570 Partial derivatives 10 When u = x2 + Y2 - z2, the graph of the equation u = 0 is a cone having its vertex at the origin. If ...

11.2 Increments, chain rule, and gradients 571 at the point (xo,yo,zo), and find the equation of the plane tangent to the latter ...

572 Partial derivatives Lagrange multiplier X should be used in the following way. When we want to find numbers x, y, z for whic ...

11.2 Increments, chain rule, and gradients 573 The extrema (if any) occur when x, y, X are such that aw ax =2x - 2A = 0 Ow ay =2 ...

574 Partial derivatives continuous. When we proved Theorem 11.23, we knew about the mean-value theorem and had enough hypotheses ...

11.2 Increments, chain rule, and gradients 575 Po is obtained by putting sf = 00 in (5), differentiating with respect to 0, and ...

576 Partial derivatives 11.3 Formulas involving partial derivatives problems require us to learn more about partial derivatives ...

11.3 Formulas involving partial derivatives 577 when y = y(x). In this case, x is both a "primary variable" and a "secondary var ...

(^578) Partial derivatives can tell us that J is a function of the independent variables x and y, the "fixed variables" being di ...

11.3 Formulas involving partial derivatives 579 Teachers and textbooks providing instruction in more advanced mathe- matics like ...

580 Partial derivatives ax/0a, ay/aa, az/aa in terms of partial derivatives of fi, f2, fa. Partial ans.: The first equation can ...

11.3 Formulas involving partial derivatives 581 Tau__au au sin 4, apcos 0 -a4, P On au au cos aY = aP sin } To P Then show that ...

582 Partial derivatives These formulas and the simple formula (4) 1 a2u 1 a2u p2 a4)_ r2 sin2 34)2 enable us to transform import ...

11.3 Formulas involving partial derivatives 583 (6) Implicit function theorem Let G be a given function of two variables x and y ...

584 Partial derivatives lar region R of Figure 11.391 is a subset of the given rectangular region RI and, moreover, (14) G(x,yi) ...

11.3 Formulas involving partial derivatives 585 whenever jOxj < S. Since G(x + Ax, y) is an increasing function ofy and G(x + ...