By Karen Pao, Frederick Soon

ISBN-10: 0387979751

ISBN-13: 9780387979755

Designed as a better half to *Basic Multivariable Calculus* by means of Marsden, Tromba, and Weinstein. This ebook parallels the textbook and reinforces the suggestions brought there with workouts, examine tricks, and quizzes. particular suggestions to difficulties and ridicule examinations also are integrated.

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**Example text**

Sections. These are intersections of graphs with a plane. Usually, the most helpful sections for graphs in ]R3 IR3 are the intersections with the = constant and y = = constant. planes x = 7. Graphing in ]R3. drawaa graph in]R3 JR 3 . The best way to draw in JR3 is to draw level =constant. constant. Then lift or drop the curves to the appropriate curves for z = "height" for z = constant. Analyzing the sections helps complete the graph. It is a good idea to review how to sketch the graph of an ellipse, a hyperbola, a cirde, circle, and a parabola from YOUf your calculus or precalculus text.

The sketch shows the level surfaces for c = = 0, -1, and -4. -4. X z 17. 21. \' X 25. )')' 17. The equation may maybe be thought of as z = iyl. Iyl. " We sketch the section in the yz-plane and translate the section parallel to the x axis. This gives us the sketch shown. 21. We note that z is missing from the equation, so we have a cylinder. cylinder. In the xy plane, the level curve is the ellipse centered at the origin, with x-intercepts at x = ±2 and y-intercepts at y = ±4. By shifting this level curve parallel to the z axis, we obtain the elliptical cylinder shown.

AFjay =f. 0, we can use the O. If 8F/8y of x as long as 8F/8y formula dy/dx dyfdx = = -Fx/F -Fx/Fyy = -3/(2y + 1). This doesn't look like what we calculated in part (a), Ca), but if we substitute y v'-3 y-3 -12x)/2, we get (-1 ± V-3 ~ dx (-1 -3 (-1 -12X)) (-1 ± )-3 J-3 -12x)) ";-3 3 ";-3 v'-3 -12x = ~---:---:-~;::=:;::=~~---=---:-~:===:;:::==~- = ±t=~~ ±-;=~~ 2 2 +1 21. Let F(x, y, z) = 0. O. The chain rule teIls tells us that 8F aF . 8x ax 8x ax 8x ax + 8F aF 8y ay . 8y ay 8x ax + 8F aF . 8z az 8z az 8x ax = o.

### Student's Guide to Basic Multivariable Calculus by Karen Pao, Frederick Soon

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