Green's theorem is a fundamental theorem in vector calculus that relates the scalar products of linear forms on two vector spaces. It states that if V and W are...
Green's theorem is a fundamental theorem in vector calculus that relates the scalar products of linear forms on two vector spaces. It states that if V and W are two vector spaces, and f:V->W and g:W->V are linear forms, then the following equation holds:
for all vectors v in V.
In simpler words, this means that the scalar product of two linear forms is equal to the product of the individual scalar products.
Here's a more formal proof of Green's theorem:
Let f:V->W and g:W->V be linear forms. Then, by the linearity of the scalar product, we have the following:
for all vectors v in V.
Since this equation holds for all vectors v in V, we have the following:
for all vectors v in V