Question

Show that each of the vector fields F=yi+x,j. G=y/x²+y i+-x/x²+y² j, and H = 5x/√x²+y² j are gradient vector fields on some domain (not necessarily the whole plane) by finding a potential function for each.

For F, a potential function is f(x, y) =
For G, a potential function is g(x, y) =
For Ĥ, a potential function is h(x, y) =

Answer 1

To prove the vector fields F, G, and H are gradient fields, potential functions f(x, y) = x^2/2 + y^2/2, g(x, y) = arctan(y/x), and h(x, y) = 5arctan(y/x) are found, respectively, satisfying the relationship ∇f = F, ∇g = G, and ∇h = H within their specific domains.

Step-by-step explanation:

To show that the vector fields F, G, and H are gradient vector fields, we need to find the potential functions f(x,y), g(x,y), and h(x,y), respectively, that satisfy the property ∇f = F, ∇g = G, and ∇h = H. A gradient vector field is one where the vector field can be expressed as the gradient of some scalar potential function.

Finding Potential Functions

Vector Field F

Given F = yi + xj, to find a potential function f(x,y), we integrate the x-component with respect to x holding y constant and the y-component with respect to y holding x constant, and combine any functions of the other variable:

• ∫ (y)dy = y2/2 + Cx(x)
• ∫ (x)dx = x2/2 + Cy(y)

Therefore, the potential function f(x, y) = x2/2 + y2/2 satisfies ∇f = F.

Vector Field G

For G = y/x2 + y i - x/x2 + y2 j, we integrate similarly:

• ∫ (y/x2 + y)dx = arctan(y/x) + Cy(y)
• ∫ (-x/x2 + y2)dy = -arctan(x/y) + Cx(x)

If we choose the constant of integration appropriately, g(x, y) = arctan(y/x) is a potential function that works for G.

Vector Field H

H = 5x/√x2+y2 j requires integrating with respect to y:

• ∫ (5x/√x2+y2)dy = 5arctan(y/x) + C(x)

The potential function h(x, y) = 5arctan(y/x) yields ∇h = H.

Note: For each potential function, the domain must exclude points where the denominators are zero or where the functions are not defined.