4/5 Multivariable Calculus

Graph where every point has a vector associated with it

Represent many things in nature

 $F=<M,\ N,\ P>$  

Come from a potential function

 $F=\nabla f$  

Test if conservative by cross-partial check:  $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$  

In 3 Dimensions,  $F$  is conservative if  $curl\left(F\right)=0$  

 $curl\left(F\right)=\nabla\times F$  

 $div\left(F\right)=\nabla\cdot F$  

Two Types, but many forms

For Functions:  $\int_C^{ }f\left(x,y\right)\ ds\ =\ \int_C^{ }f\left(x\left(t\right),\ y\left(t\right)\right)\sqrt{\left(x'\left(t\right)\right)^2+\left(y'\left(t\right)\right)^2}dt$  

For a Vector Field: $\int_C^{ }F\ dr\ =\ \int_C^{ }F\cdot T\ ds\ =\ \int_C^{ }M\ dx\ +N\ dy$  

If $F$  is a conservative vector field, then  $\int_C^{ }F\cdot\ dr\ =\ \int_a^b\nabla f\cdot\ dr=f\left(b\right)-f\left(a\right)$  where C is a path that goes from a point a to a point b

 $F$  is conservative

 $F$  is independent of path

 $\oint_C^{ }F\ dr\ =\ 0$