By Karen Pao, Frederick Soon
Designed as a better half to Basic Multivariable Calculus by way of Marsden, Tromba, and Weinstein. This e-book parallels the textbook and reinforces the options brought there with routines, research tricks, and quizzes. exact strategies to difficulties and ridicule examinations also are incorporated.
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Example text
The The area area of of the the parallelogram parallelogram with those those two two vectors vectors as with as adjacent adjacent sides sides isis the the magnitude magnitude of of Vl Vt X X V2 V2 = = ii jj kk 11 11 00 =-k. = -k. 2 2 1 1 00 Since IIVl llvt xx v211 v2ll == 1, Since 1, the the area area of of the the triangle triangle is is 1/2. 1/2. 45. A vector orthogonal to both 3i + 2k and j - kk is (3i + 2k) xX (j - k) = (j- ii 3 o0 jj kk 0 2 = -2i + 3j + 3k. 1 -1 The length of the cross product is v' y'4 + 9 + 9 = y'22, ¥'22, so we can normalize to get the desired unit vector maHze 1 .
Differentiation Differentiation 2. Writing this this in in terms terms of of t,t, we we get get Writing dTI/dt dt dT 16(3 ++ 8t) 8t) cos cos (3 (3 -- 2t) 2t) -- 8(38(3 - 2t) 2t)22 cos cos (3 (3 ++ 8t) 8t) == 16(3 +2(3 ++ 8t) 8t)22 sin sin (3(3 - 2t) 2t) ++ 4(34(3 - 2t) 2t) sin sin (3 (3 ++ 8t). 8t). +2(3 To compute compute dT dTI/ dt dt directly, directly, first first note note that that To T(t) T(t) (3 ++ 8t) 8t)22 cos cos (3(3 - 2t) 2t) -- (3(3 - 2t) 2t)22 sin sin (3 (3 ++ 8t). 8t). = (3 Thus, by by one-variable one-variable calculus calculus methods, methods, Thus, dTI/ dt 8t)22 sin (3(3 - 2t) = 16(3 + 8t) cos (3 - 2t) + 2(3 + 8t) +4(3 - 2t) sin (3 + 8t) - 8(38(3 - 2t) 2t)22 cos (3 + 8t).
O. 0. · (x f(x) == /(Xo). We compute V I(x) normal to the tangent plane, which is V I(Xo). v'3, 1). Therefore, the desired 4v'3, 6) at (1, V3, (2x, 4y, 6z), which is (2, 4V3, 2x+4v'3y+ v'3) +6(z -1) == 0 or 2x+4V3y+ +4v'3(y- V3) tangent plane is 2(x-l) 2(x-1) +4V3(y10. + 3z == 10. 5Gradients Gradientsand andDirectional DirectionalDerivatives Derivatives 39 39 37. The The equation equation of ofaa tangent 37. 0. For For this this particular particular ease, case, we we eompute compute Vf(x, f(x, y) y) == (2x,4y), (2x, 4y), which V which isis 2i 2i + + 4j4j atat the the point point (1,1).