Explanation Given information: The vector F = 〈 z − y 2 , x + z 3 , y + x 2 〉 has been given. Formula used: The divergence of a vector F = 〈 F 1 , F 2 , F 3 〉 can be obtained by partially integration with respect to the variable. div ( F ) = ∂ F 1 ∂ x + ∂ F 2 ∂ y + ∂ F 3 ∂ z The curl od a vector can be obtained by the symbolic determinant of the following vector. curl ( F ) = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z F 1 F 2 F 3 | Calculation: First, F 1 , F 2 and F 3 should be identified from the given vector F = 〈 z − y 2 , x + z 3 , y + x 2 〉 . F 1 = z − y 2 F 2 = x + z 3 F 3 = y + x 2 Then by substituting these values into the formula of div ( F ) . div ( F ) = ∂ ( z − y 2 ) ∂ x + ∂ ( x + z 3 ) ∂ y + ∂ ( y + x 2 ) ∂ z = 0 + 0 + 0 div ( F ) = 0 Then, we should build up the matrix that denotes the vector F

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

See similar textbooks

Videos

Question

Chapter 17.1, Problem 27E

To determine

To find:

The div(F) and curl(F) of the given vector F.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 17.1, Problem 26E

Chapter 17.1, Problem 28E

Students have asked these similar questions

(12 points) Let E={(x, y, z)|x²+ y² + z² ≤ 4, x, y, z > 0}. (a) (4 points) Describe the region E using spherical coordinates, that is, find p, 0, and such that (x, y, z) (psin cos 0, psin sin 0, p cos) € E. (b) (8 points) Calculate the integral E xyz dV using spherical coordinates.

(10 points) Let f(x, y, z) = ze²²+y². Let E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z < 3}. Calculate the integral y, f(x, y, z) dV.

(14 points) Let f: R3 R and T: R3. →R³ be defined by f(x, y, z) = ln(x²+ y²+2²), T(p, 0,4)=(psin cos 0, psin sin, pcos). (a) (4 points) Write out the composition g(p, 0, 4) = (foT)(p,, ) explicitly. Then calculate the gradient Vg directly, i.e. without using the chain rule. (b) (4 points) Calculate the gradient Vf(x, y, z) where (x, y, z) = T(p, 0,4). (c) (6 points) Calculate the derivative matrix DT(p, 0, p). Then use the Chain Rule to calculate Vg(r,0,4).

Chapter 17 Solutions

Calculus - Standalone book

Ch. 17.1 - Prob. 1PQ Ch. 17.1 - Prob. 2PQ Ch. 17.1 - Prob. 3PQ Ch. 17.1 - Prob. 4PQ Ch. 17.1 - Prob. 1E Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The div ( F ) and curl ( F ) of the given vector F .

Solution Summary: The author explains that the divergence and the curl of a vector F can be obtained by the symbolic determinant of the following vector.

Videos

To find:

The div(F) and curl(F) of the given vector F.

Want to see the full answer?

Chapter 17 Solutions

The div ( F ) and curl ( F ) of the given vector F .

Solution Summary: The author explains that the divergence and the curl of a vector F can be obtained by the symbolic determinant of the following vector.

Videos

To find: The div(F) and curl(F) of the given vector F.

Want to see the full answer?

Chapter 17 Solutions

To find:

The div(F) and curl(F) of the given vector F.