vector fields {cal,exm3} TO REVIEW

Why do we care about conserative vect fields?

FUndamental therom of line $\int$

• Vect Line $\int$: ${\int }_{C}v\cdot ds$ = ${\int }_b^{b}v\cdot r^{\prime }(t)dt$
  • How is r related to C?

Q: Find $\Phi$

$$\nabla \Phi =V=<2xy+y,x^2+x-1>,\Phi =?$$

• Identity partials: (where C(i),D = constants)
  • ${\Phi }_x=<\frac{x^2}{2}+C(y)+D>$
  • ${\Phi }_y=<xy+B(x)+D>$
  • $\Phi =<xy+\frac{x^2}{2}+D>$
    • $\nabla \Phi =<y+x,x>$ → Not conservative! End early :)

Suppose have $V(x,y)=<S(x,y),T(x,y)>$

• VF conserative if
  • $V=\nabla \varphi$
  • In this sense we have$\Phi (x,y)=x^{2}y+y\nabla \Phi =<2xy,x^2+1>$
    • Which is split apart dependent on what avrs they are split upon
    • Hence if we have$$$$

Vector fields!! (THIS WILL BE ON THE EXAM!)

lamr edu vector fields

• Finding if func is conserative? lruc
• $V(x,y)=<2x,2y>$
  • Then find $S(x,y)=2x,T(x,y)=2y$
  • Then the partials of each is 0
• If $V(x,y)=\nabla af(x,y)$ what is f?
• $f(x,y)=A(x,y)+B(x)+C(y)+D$, where D is some const
• $f_x=S=2x$ Hence $f(x,y)=\int 2xdx=x^2+(C(y)+D)$
• $f_y=2y$ then $f(x,y)=\int 2ydy=y^2+(B(x)+D)$
• $x^2+C(y)+D=y^2+B(x)+D$
  • Hence $\boxed{f(x,y)=x^2+y^2+D}$