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# Theorems Concerning Vector Fields

### Theorem

Given the vector field $\stackrel{\to }{E}\left(x,y,z\right)=f\left(x,y,z\right)\stackrel{\to }{i}+g\left(x,y,z\right)\stackrel{\to }{j}+h\left(x,y,z\right)\stackrel{\to }{k}$ . If $\stackrel{\to }{E}=-\nabla \phi$ , then:
1. $\stackrel{\to }{E}$ is a conservative field.
2. $\nabla \phi$ is the gradient field of $\stackrel{\to }{E}$ .
3. $\phi$ is the potential of $\stackrel{\to }{E}$ .

### The Fundamental Theorem of Line Integrals

Given a vector field $\stackrel{\to }{E}\left(x,y\right)=f\left(x,y\right)\stackrel{\to }{i}+g\left(x,y\right)\stackrel{\to }{j}$ ,where f and g are continuous functions on plane D, which contains the points $\left({x}_{0},{y}_{0}\right)$ and $\left({x}_{1},{y}_{1}\right)$ . If $\stackrel{\to }{E}\left(x,y\right)=-\nabla \phi \left(x,y\right)$ for every point in D, then for each smooth curve C in D, which start at $\left({x}_{0},{y}_{0}\right)$ and ends at $\left({x}_{1},{y}_{1}\right)$ , the following is true: ${\int }_{c}\stackrel{\to }{E}\left(x,y\right)·d\stackrel{\to }{r}=\phi \left({x}_{1},{y}_{1}\right)-\phi \left({x}_{0},{y}_{0}\right)$ .

### Equivalance Theorem

Given a vector field, $\stackrel{\to }{E}\left(x,y\right)=f\left(x,y\right)\stackrel{\to }{i}+g\left(x,y\right)\stackrel{\to }{j}$ , in which f and g are continous functions on a plane D, then the following three statements are equivalent (either they are all true or non of them is true):
1. $\stackrel{\to }{E}$ is a conservative field in D.
2. ${\oint }_{C}\stackrel{\to }{E}·d\stackrel{\to }{r}=0$ for each closed curve C in D.
3. ${\oint }_{C}\stackrel{\to }{E}·d\stackrel{\to }{r}$ does not depend on the path, for each curve C in D. 