Final answer:
The vector field F(x, y) = (10x + 3y)i + (3x + 10j)j is conservative.
The potential function f(x, y) = 5x^2 + 6xy + 5y^2.
Step-by-step explanation:
A vector field is said to be conservative if its curl is zero.
To determine if the vector field F(x, y) = (10x + 3y)i + (3x + 10j)j is conservative or not, we need to compute its curl.
The curl of a vector field is given by:
curl F = (∂Fy/∂x - ∂Fx/∂y)k
where k is the unit vector in the z-direction.
For the given vector field F, we have:
∂Fx/∂y = 3
∂Fy/∂x = 3
Therefore, curl F = (3 - 3)k = 0
Since the curl is zero, the vector field F is conservative.
To find the potential function f, we need to integrate each component of F(x, y) with respect to its corresponding variable.
So we have:
f(x, y) = ∫(10x + 3y)dx
= 5x^2 + 3xy + C(y)
f(x, y) = ∫(3x + 10y)dy
= 3xy + 5y^2 + C(x)
Since the potential function has a value of zero at the origin, we can set C(x) = C(y) = 0.
Therefore, f(x, y) = 5x^2 + 3xy + 3xy + 5y^2
= 5x^2 + 6xy + 5y^2.