Calculus
7th Edition
ISBN: 9781524916817
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
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_forwarda) What is the flux of the constant vector field F = (a, b, c) through any closed surface? b) Is the flux of F = (x³, y³, z³) through any closed surface positive, negative, or zero?arrow_forwardCalculate the cross product assuming that u x v = (4, 3, 0). v x (u + v)arrow_forward
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- Let F(x, y, z) = xzi+yzj + xyk, Surface S is the part of the sphere x² + y² + z² = 4 that lies inside the cylinder x² + y² = 1 and above the xy-plane. Find a vector equation of the boundary curve C. (b) Compute for FdR (c) Compute for C F.Tdsarrow_forwardU (x, y) 4c(y + 1)i + xyj, and V(x, y) = Consider two vector fields in the xy plane, given in the Cartesian coordinates as: cyi - xj, where c is a constant. Find where in the xy plane the vectors of these two fields are parallel to one another, and where they are mutually orthogonal. =arrow_forwardSet V = {(x, y) | xER, YER}. Define addition of vectors by the rule (x, y) + (x, y) = (y + y',x+x') and define scalar multiplication by the rule (x,y) c(x, y) = (y,x) (0,0) if c>0 if c < 0 if c = 0 For example, (1, 2) + (3, 4) = (6,4) and -2(3, 4) =(4,3). For each of the vector space axioms (A1)-(A4), (M1)-(M4), either prove that the axiom holds or prove that it does not hold. Use the axioms as described in the lecture notes. Here are a few things to keep in mind. • To disprove (A2), you would need to show that no vector z = (a, b) has the property that for all u = (x, y), u + z = u and z+u = u. • If (A2) is false, (A3) is automatically false as it relies on the existence of a zero vector. • To disprove (A3), you would need to give a specific example of a vector u such that for all other vectors wat least one of u + w = z or w+u = z is false.arrow_forward
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