Question

3. Consider the vector field F(x, y, z) = <2xy^3 + z^2, 3x^2y^2 + 2yz, y^2 + 2xz>. (a) Show that F is a conservative vector field and find a potential function f. (b) Evaluate the line integral ?_C F · dr, where C is any path from (0, 1, -1) to (1, 2, 1).

          3. Consider the vector field F(x, y, z) = <2xy^3 + z^2, 3x^2y^2 + 2yz, y^2 + 2xz>.
(a) Show that F is a conservative vector field and find a potential function f.
(b) Evaluate the line integral
?_C F · dr,
where C is any path from (0, 1, -1) to (1, 2, 1).

3. Consider the vector field F(x, y, z) = <2xy^3 + z^2, 3x^2y^2 + 2yz, y^2 + 2xz>.
(a) Show that F is a conservative vector field and find a potential function f.
(b) Evaluate the line integral
?C F · dr,
where C is any path from (0, 1, -1) to (1, 2, 1).

Added by Dylan G.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

08/29/2023

Step 1

The curl of F is given by: curl(F) = (∂Fz/∂y - ∂Fy/∂z, ∂Fx/∂z - ∂Fz/∂x, ∂Fy/∂x - ∂Fx/∂y) Calculating the partial derivatives, we have: ∂Fz/∂y = 2y ∂Fy/∂z = 2 ∂Fx/∂z = 2z ∂Fz/∂x = 0 ∂Fy/∂x = 6xy^2 ∂Fx/∂y = 6xy^2 Show more…

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3. Consider the vector field F(x, y, z) = <2xy^3 + z^2, 3x^2y^2 + 2yz, y^2 + 2xz> (a) Show that F is a conservative vector field and find a potential function f. (b) Evaluate the line integral ∫_C F · dr, where C is any path from (0, 1, -1) to (1, 2, 1).