Explanation Concept used: The divergence theorem states that, Let S be a smooth, orientable surface that encloses a solid region R in ℝ 3 . If F → is a continuous vector field whose components have continuous partial derivatives in an open set containing R , then ∬ S F → ⋅ N → d s = ∭ R div F → d V ; where N → is the outward unit normal field for the surface S . Consider the vector function F → = 3 x i ^ + 5 y j ^ + 6 z k ^ and the tetrahedron bounded by the coordinate planes and the plane D : 2 x + y + z = 4 . Here, ∬ S F → ⋅ N → d s = ∬ D 〈 3 x , 5 y , 6 z 〉 ⋅ 〈 − ( − x 4 − 2 x − y ) , − ( − y 4 − 2 x − y ) , 1 〉 d A = ∬ D [ 3 x 2 4 − 2 x − y + 5 y 2 4 − 2 x − y + 6 ( 4 − 2 x − y ) ] d A = ∫ 0 2 π ∫ 0 4 [ 3 r 2 cos 2

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

See similar textbooks

Question

Chapter 13.7, Problem 10PS

To determine

To verify: The divergence theorem for the vector function F→=3xi^+5yj^+6zk^ and the tetrahedron bounded by the coordinate planes and the plane D:2x+y+z=4 .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.7, Problem 9PS

Chapter 13.7, Problem 11PS

Students have asked these similar questions

Suppose that the graph below is the graph of f'(x), the derivative of f(x). Find the locations of all relative extrema, and tell whether each extremum is a relative maximum or minimum. Af'(x) Select the correct choice below and fill in the answer box(es) within your choice. (Simplify your answer. Use a comma to separate answers as needed.) -10 86-4-2 -9- B 10 X G A. The function f(x) has a relative maximum at x= relative minimum at x = and a B. The function f(x) has a relative maximum at x= no relative minimum. and has C. There is not enough information given. D. The function f(x) has a relative minimum at x= no relative maximum. and has E. The function f(x) has no relative extrema.

K Find the x-values of all points where the function has any relative extrema. Find the value(s) of any relative extrema. f(x) = 12x+13x 12/13 Select the correct choice below and, if necessary, fill in any answer boxes within your choice. OA. There are no relative maxima. The function has a relative minimum of (Use a comma to separate answers as needed.) OB. There are no relative minima. The function has a relative maximum of (Use a comma to separate answers as needed.) OC. The function has a relative maximum of at x= (Use a comma to separate answers as needed.) OD. There are no relative extrema. at x= at x= and a relative minimum of at x=

K Find the x-values of all points where the function has any relative extrema. Find the value(s) of any relative extrema. f(x) = - 2 3 9 -4x+17 Select the correct choice below and, if necessary, fill in any answer boxes within your choice. OA. There are no relative minima. The function has a relative maximum of (Use a comma to separate answers as needed.) OB. There are no relative maxima. The function has a relative minimum of (Use a comma to separate answers as needed.) OC. The function has a relative maximum of at x= (Use a comma to separate answers as needed.) OD. There are no relative extrema. at x= at x= and a relative minimum of at x=

Chapter 13 Solutions

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

Ch. 13.1 - Prob. 1PS Ch. 13.1 - Prob. 2PS Ch. 13.1 - Prob. 3PS Ch. 13.1 - Prob. 4PS Ch. 13.1 - Prob. 5PS Ch. 13.1 - Prob. 6PS Ch. 13.1 - Prob. 7PS Ch. 13.1 - Prob. 8PS Ch. 13.1 - Prob. 9PS Ch. 13.1 - Prob. 10PS

Knowledge Booster

To verify: The divergence theorem for the vector function F → = 3 x i ^ + 5 y j ^ + 6 z k ^ and the tetrahedron bounded by the coordinate planes and the plane D : 2 x + y + z = 4 .

Solution Summary: The author explains the divergence theorem for the vector function stackrelto and the tetrahedron bounded by the coordinate planes.

To verify: The divergence theorem for the vector function F→=3xi^+5yj^+6zk^ and the tetrahedron bounded by the coordinate planes and the plane D:2x+y+z=4 .

Want to see the full answer?

Chapter 13 Solutions