Please help with this Calculus problem

Question

Consider the vector field F= (7x+4y,4x+5)

Is this vector field conservative?

if so: Find a function f so that F=∇f

f(x,y) = _____ + K

Answer 1

To determine whether the vector field F is conservative, we need to check if it satisfies the condition that the curl of F is zero. That is,

curl(F) = ∂Q/∂x - ∂P/∂y = 0

where F = (P, Q).

Here, we have P = 7x + 4y and Q = 4x + 5. Therefore,

∂Q/∂x = 4 and ∂P/∂y = 4

Since these partial derivatives are equal, the curl of F is zero, and the vector field F is conservative.

To find a potential function f, we need to find a function whose gradient is equal to F. That is,

∇f = (7x+4y, 4x+5)

Integrating the first component with respect to x gives:

f(x,y) = 7/2 x^2 + 4xy + g(y)

where g(y) is a constant of integration with respect to x.

Taking the partial derivative of f with respect to y gives:

∂f/∂y = 4x + g'(y)

Comparing this with the second component of F, we have:

g'(y) = 5

Integrating with respect to y gives:

g(y) = 5y + K

where K is a constant of integration. Therefore, the potential function f is:

f(x,y) = 7/2 x^2 + 4xy + 5y + K

where K is an arbitrary constant. Thus, we have found a potential function f for the vector field F, and it is given by the equation above.

Answer 2

curl F= 0 thus F is conservative ... note that fx=7x+4y by definition, thus f = 3.5x²+4yx+h(y) where h(y) is a constant of integration. note further that fy=4x+5 by definition and fy=4x+h'(y) (take partial of f from above). Integrate fy wrt y to find f=4xy+h. Set the two versions of f = to each other and solve for h'(y)=5. Integrate to find h(y)=5y+C thus f(x,y)=3.5x²+4xy+5y+C.

Answer 3

To determine whether the vector field F= (7x+4y, 4x+5) is conservative or not, we need to calculate its curl. If the curl is zero, then the vector field is conservative.

So, let's calculate the curl of F:

∇ x F = (∂/∂x, ∂/∂y, ∂/∂z) x (7x+4y, 4x+5, 0) = (∂(4x+5)/∂y - ∂(7x+4y)/∂z, ∂/∂z(7x+4y) - ∂/∂x(4x+5), ∂/∂x(4x+5) - ∂/∂y(7x+4y)) = (0, 0, -3)

Since the z-component of the curl is not zero, the vector field is not conservative.

Therefore, there is no function f such that F=∇f.

We cannot find a function f(x,y) such that F=∇f.

So the final answer is:

f(x,y) = N/A (since F is not conservative) + K, where K is an arbitrary constant.