Question

Consider F and C below. F(x, y) = 6xy2 i + 6x2y j C: r(t) = t + sin 1 2 πt , t + cos 1 2 πt , 0 ≤ t ≤ 1 (a) Find a function f such that F = ∇f. f(x, y) = (b) Use part (a) to evaluate C ∇f · dr along the given curve C.

Answer 1

a. If $\vec F$ is a gradient field of a scalar function $f$, then

$\dfrac{\partial f}{\partial x}=6xy^2$

$\dfrac{\partial f}{\partial y}=6x^2y$

From these equations we can find

$f(x,y)=3x^2y^2+g(y)$

$\dfrac{\partial f}{\partial y}=6x^2y=6x^2y+\dfrac{\mathrm dg}{\mathrm dy}\implies\dfrac{\mathrm dg}{\mathrm dy}=0\implies g(y)=C$

$\implies\boxed{f(x,y)=3x^2y^2+C}$

b. By the gradient theorem (i.e. fundamental theorem of calculus for line integrals),

$\displaystyle\int_C\nabla f\cdot\mathrm dr=f(\vec r(1))-f(\vec r(0))=\boxed{12}$