# SOLVED: Consider the following region R and the vector field. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in the flux form of Green’s Theorem and check for c

Lec 23: Flux; normal form of Green’s theorem | MIT 18.02 Multivariable Calculus, Fall 2007
Lec 23: Flux; normal form of Green’s theorem | MIT 18.02 Multivariable Calculus, Fall 2007

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Consider the following region R and the vector field. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in the flux form of Green’s Theorem and check for consistency: State whether the vector field is source-free.
(2xyx^2 – y^2); R is the region bounded by y = x(2 – x) and y = 0.
a. The two-dimensional divergence is
b. Set up the integral over the region:
âˆ«âˆ«(2xyx^2 – y^2) dy dx
Set up the line integral for the y = x(2 – x) boundary.

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04:11

Consider the following region R and the vector field F_a: Compute the two-dimensional divergence of the vector field. b: Evaluate both integrals in the flux form of Green’s Theorem and check for consistency. c: State whether the vector field is source-free.

F = (8xy, x^2 – 4y^2); R is the region bounded by y = x(4 – x) and y = 0.

The two-dimensional divergence is

âˆ‡ Â· F = âˆ‚(8xy)/âˆ‚x + âˆ‚(x^2 – 4y^2)/âˆ‚y

Set up the integral over the region.

âˆ«âˆ«R âˆ‡ Â· F dA

dy dx

05:32

17.4.29 – Setup & Solve

Consider the following region R and the vector field F. Compute the two-dimensional divergence of the vector field.

F = (2xy, x^2 – y)

R is the region bounded by y = x(2 – x) and y = 0.

a. The two-dimensional divergence is

b. Set up the integral over the region:

âˆ«âˆ«R 2x(2 – x) dy dx

Set up the line integral for the y = x(2 – x) boundary:

âˆ«dt

02:43

Consider the following region R and the vector field F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in Green’s Theorem and check for consistency:

F = (-x, -y); R: x + y^2 < 9

03:08

Consider the following region R and the vector field F. Compute the two-dimensional divergence of the vector field. b: Evaluate both integrals in Green’s Theorem and check for consistency: F = (Zy, 4x); R is the triangle with vertices (0,0), (4,0), and (0,4): a. The two-dimensional divergence isb. Set up the integral over the region:âˆ«âˆ« F Â· dA(Type exact answers)

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