Compute the line integral of the vector field F(x, y, z) = ⟨−y, x, z⟩ along...

Compute the line integral of the vector field F(x, y, z) = ⟨−y, x, z⟩ along the curve which is given by the intersection of the cylinder x 2 + y 2 = 4 and the plane x + y + z = 2 starting from the point (2, 0, 0) and ending at the point (0, 2, 0) with the counterclockwise orientation.

Expert Solution

Colby Messinger answered 2 years ago

Compute the derivative of the given vector field F. Evaluate the line integral of F(x,y,z) = (y+z+yz , x+z+xz , x+y+xy )

Compute the derivative of the given vector field F. Evaluate the line integral of F(x,y,z) = (y+z+yz , x+z+xz , x+y+xy )over the path C consisting of line segments joining (1,1,1) to (1,1,2), (1, 1, 2) to (1, 3, 2), and (1, 3, 2) to (4, 3, 2) in 3 different ways, along the given path, along the line from (1,1,1) to (4,3,2), and finally by finding an anti-derivative, f, for F.

Using Green’s theorem, compute the line integral of the vector field below, along the curve x^2...

Using Green’s theorem, compute the line integral of the vector field below, along the curve x^2 - 2x + y^2 = 0 , with the counterclockwise orientation. Don’t compute the FINAL TRIG integral. F(x,y) = < (-y^3 / 3) - cos(x^7) , cos(y^9 + y^5) + (x^3 / 3) > .

Using Green’s theorem, compute the line integral of the vector field below, along the curve x^2-2x+...

Using Green’s theorem, compute the line integral of the vector field below, along the curve x^2-2x+ y^2=0 , with the counterclockwise orientation. Don’t compute the FINAL TRIG integral. F(x,y)=<- y^3/3-cos⁡(x^7 ) ,cos(y^9+y^5 )+ x^3/3> .

Verify the Divergence Theorem for the vector field F(x, y, z) = < y, x ,...

Verify the Divergence Theorem for the vector field F(x, y, z) = < y, x , z^2 > on the region E bounded by the planes y + z = 2, z = 0 and the cylinder x^2 + y^2 = 1. By Surface Integral: By Triple Integral:

The flow of a vector field is F=(x-y)i+(x^2-y)j along the straight line C from the origin...

The flow of a vector field is F=(x-y)i+(x^2-y)j along the straight line C from the origin to the point (3/5, -4/5) A. Express the flow described above as a single variable integral. B. Then compute the flow using the expression found in part A. Please show all work.

Let f(x,y) be a scalar function, and let F(x,y,z) be a vector field. Only one of...

Let f(x,y) be a scalar function, and let F(x,y,z) be a vector field. Only one of the following expressions is meaningful. Which one? a) grad f x div F b) div(curl(grad f)) c) div(div F) d) curl(div(grad f)) e) grad(curl F)

Evaluate the line integral ∫_C F⋅dr, where F(x,y,z)=−4sin(x)i+3cos(y)j−4xzk and C is given by the vector function...

Evaluate the line integral ∫_C F⋅dr, where F(x,y,z)=−4sin(x)i+3cos(y)j−4xzk and C is given by the vector function r(t)=t^6i−t^5j+t^4k , 0≤t≤1.

Use the extended divergence theorem to compute the total flux of the vector field F(x, y,...

Use the extended divergence theorem to compute the total flux of the vector field F(x, y, z) = −3x2 + 3xz − y, 2y3 − 6y, 9x2 + 4z2 − 3x outward from the region F that lies inside the sphere x2 + y2 + z2 = 25 and outside the solid cylinder x2 + y2 = 4 with top at z = 1 and bottom at z = −1.

3. a) Consider the vector field F(x, y, z) = (2xy2 z, 2x 2 yz, x2...

3. a) Consider the vector field F(x, y, z) = (2xy2 z, 2x 2 yz, x2 y 2 ) and the curve r(t) = (sin t,sin t cost, cost) on the interval [ π 4 , 3π 4 ]. Calculate R C F · dr using the definition of the line integral. [5] b) Find a function f : R 3 → R so that F = ∇f. [5] c) Verify your answer from (a) using (b) and the Fundamental...

Given the line integral ∫c F(r) · dr where F(x, y, z) = [mxy − z3...

Given the line integral ∫c F(r) · dr where F(x, y, z) = [mxy − z3 ,(m − 2)x2 ,(1 − m)xz2 ] (a) Find m such that the line integral is path independent; (b) Find a scalar function f such that F = grad f; (c) Find the work done in moving a particle from A : (1, 2, −3) to B : (1, −4, 2).

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