Question

The Vector Field f(x, y) = (2x + 2y^2)i + (4xy - 6y^2)j has exactly one potential function f (x, y) that satisfies f(0, 0). Find this potential function , then find the value of this potential function at the point (1, 1).

Answer 1

Step 1)

we have,

Now,

And,

Step 2)

we know that,

    -------------------------------------------------1)

we can write,

equate f_y = 4xy + h'(y) with Q = 4xy - 6y² we can write,

Hence,

Put value of h(y) in equation 1) we can write,

Step 3)

we have,

    ------------------------------------------2)

which is nothing but the potential function for the given vector field

Potential function satisfies f(0,0) = 0 so put x = 0, y = 0 and f(0,0) = 0 in equation 2) we can write,

Put C = 0 in equation 2) we have,

we have to find f(1,1) so put x = 1 and y = 1 we can say that,