Proof of Line Integral Using ∇f & ∇g: R Region, C Curve

[charmmy]

Homework Statement

Let f(x,y) and g(x,y) be continuously differentiable real-valued functions in a region R. Show that ∫f ∇g · dr ]= − ∫g ∇f · dr for any closed curve C in R.

Homework Equations

The Attempt at a Solution

I don't really know where to start, so I tried to evaluate the LHS of the equation but how do I do this symbolically? and where do I lead on from ther?

 

[gabbagabbahey]

Homework Statement

Let f(x,y) and g(x,y) be continuously differentiable real-valued functions in a region R. Show that ∫f ∇g · dr ]= − ∫g ∇f · dr for any closed curve C in R.

Homework Equations

The Attempt at a Solution

I don't really know where to start, so I tried to evaluate the LHS of the equation but how do I do this symbolically? and where do I lead on from ther?

Hint: What is $\mathbf{\nabla}\left[f(x,y)g(x,y)\right]$? What is the line integral of the gradient of a function over a closed curve?

 

[charmmy]

so is LaTeX Code: \\mathbf{\\nabla}\\left[f(x,y)g(x,y)\\right] =
take h= f(x,y) g(x,y)
then ∇h=dh/dx+dh/dy...

What is the line integral of the gradient of a function over a closed curve? : is this just equal to zero?

i'm quite confused

 

[gabbagabbahey]

so is LaTeX Code: \\mathbf{\\nabla}\\left[f(x,y)g(x,y)\\right] =
take h= f(x,y) g(x,y)
then ∇h=dh/dx+dh/dy...

Well, yes, that's the basically definition of gradient. However, it isn't all that useful to you here...there is a product rule that should be in your textbook/notes that tells you how to take the gradient of a product of two scalar functions...Use that.

What is the line integral of the gradient of a function over a closed curve? : is this just equal to zero?

Yes, this is a direct consequence of the fundamental theorem of gradients.

$$\oint \mathbf{\nabla}(fg)\cdot d\textbf{r}=0$$

You can also express the integrand in terms of $f$, $g$, $\mathbf{\nabla}f$ and $\mathbf{\nabla}g$ using the product rule I mentioned above. Doing so, then splitting up the integral and using the fact that the result is zero will allow you to show the desired result.