Some questions on Green's Theorem.

[yungman]

I am working on the derivation of Green's Theorem. I might have more question in later post. I am looking at this pdf file:

http://www.math.psu.edu/roe/230H/slides_14nov.pdf

In page 3, $\int \int_R\frac{\partial N}{\partial x} dx dy=\int_c^d\int_a^{g(y)}\frac{\partial N}{\partial x} dx dy$

How is $\int_a^{g(y)}\frac{\partial N}{\partial x} dx \;=\;N[g(y),y]\;-\;N(a,y)$ as show in page 3? I don't know the steps to arrive to this.

 

[AlephZero]

This is just the fundamental theorem of calculus, or the idea that an integral is an "antiderivative".
$$\int \frac{df(x)}{dx}\,dx =f(x) + C$$
$$\int_a^b \frac{df(x)}{dx}\,dx =f(b) - f(a)$$
When you do the "inside" integration over x, y is a constant, so N is just a function of x.

 

[yungman]

This is just the fundamental theorem of calculus, or the idea that an integral is an "antiderivative".
$$\int \frac{df(x)}{dx}\,dx =f(x) + C$$
$$\int_a^b \frac{df(x)}{dx}\,dx =f(b) - f(a)$$
When you do the "inside" integration over x, y is a constant, so N is just a function of x.

Thanks, yes I forgot about this. Just went back and studied up.