Question

Consider f and c below. f(x, y) = (7 + 8xy2)i + 8x2yj, c is the arc of the hyperbola y = 1/x from (1, 1) to 3, 1 3 (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

$\dfrac{\partial f}{\partial x}=7+8xy^2$

$\dfrac{\partial f}{\partial y}=8x^2y$

The first equation gives

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

Differentiating with respect to $y$ gives

$8x^2=8x^2y+\dfrac{\mathrm dg}{\mathrm dy}\implies\dfrac{\mathrm dg}{\mathrm dy}=0\implies g(y)=K$

for some constant $K$. So

$f(x,y)=7x+4x^2y^2+K$

and by the fundamental theorem of calculus,

$\displaystyle\int_C\nabla f\cdot\mathrm d\vec r=f\left(3,\frac13\right)-f(1,1)=25-11=\boxed{14}$