When are Line/Surface Integrals Used?

[Jacob87411]

I just want to verify how/what each of these is used for:

Fundamental Theorem for Line Integrals - This is like the regular fundamental theorem but you use the gradient of F? And this is used for curved lines

Greens Theorem - This is only used for simple enclosed curves

Stokes' Theorem & Divergence Theorem - I'm confused when you use these, are they for 3-dimensional shapes? Thanks

 

[Jheriko]

Greens Theorem - This is only used for simple enclosed curves

Stokes' Theorem & Divergence Theorem - I'm confused when you use these, are they for 3-dimensional shapes? Thanks

The Divergence Theorem let's you write a surface integral as a volume integral.

Green's theorem let's you write a line integral around a closed curve as a surface integral over the contained surface (as long as it lies in the plane).

Not sure about the fundamental theorem of line integrals...

 

[Sculptured]

The Fundamental Line Theorem is used when the vector field you are integrating the curve over is conservative (curl = 0). A conservative vector field is the gradient of some function so this theorem allows you to just find the original function and take the difference between any two points in the field to give the evaluation for the line integral of a curve between them.