Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)
6th Edition
ISBN: 9781524908102
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 13.7, Problem 44PS
To determine
To find: The value of
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Let
F =
=
(x, y, z)
(x² + y² + z²)3/2
Use the fact that V F = 0 when (x, y, z)
following surfaces.
(0,0,0) to find the flux of F through the
The square with corners (1, 1, 1), (1, −1, 1), (−1, 1, 1), and (-1, -1, 1), oriented
upward. Hint: Use the fact that this is a face of a cube centered at the origin.
4. Consider the vector function r(z, y) (r, y, r2 +2y").
(a) Re-write this vector function as surface function in the form f(1,y).
(b) Describe and draw the shape of the surface function using contour lines and algebraic analysis
as needed. Explain the contour shapes in all three orthogonal directions and explain and label
all intercepts as needed.
(c) Consider the contour of the surface function on the plane z=
for this contour in vector form.
0. Write the general equation
Let F = -9zi+ (xe#z– 2xe**)}+ 12 k. Find f, F·dĀ, and let S be the portion
of the plane 2x + 3z = 6 that lies in the first octant such that 0 < y< 4 (see
figure to the right), oriented upward.
Z
Explain why the formula F · A cannot be used to find the flux of F through the surface S. Please be
specific and use a complete sentence.
Chapter 13 Solutions
Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)
Ch. 13.1 - Prob. 1PSCh. 13.1 - Prob. 2PSCh. 13.1 - Prob. 3PSCh. 13.1 - Prob. 4PSCh. 13.1 - Prob. 5PSCh. 13.1 - Prob. 6PSCh. 13.1 - Prob. 7PSCh. 13.1 - Prob. 8PSCh. 13.1 - Prob. 9PSCh. 13.1 - Prob. 10PS
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- 2. Let U be a vector function of position in R³ with continuous second partial deriva- tives. We write (U₁, U2, U3) for the components of U. (a) Show that V (U • U) – Ü × (▼ × Ü) = (U · ▼) Ū, (V where ((UV) U), = U₁ (b) We define the vector function of position, by setting = V × . If the condition V U = 0 holds true, show that ▼ × ((Ū · V) Ű) = (Ũ · V) Ñ - (Ñ· ▼) ū. (Hint: The representation of (UV) U from the first part might be useful.)arrow_forwardCalculate the cross product assuming that u x v = (4, 3, 0). v x (u + v)arrow_forwardEvaluate the circulation of G = xyi + zj + 4yk around a square of side 4, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis. Circulation = Jo F. dr =arrow_forward
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