Surface integrals! {cal3}

Visuals

• height map
• heat map
• PLOT EVERYTHING YOU WORK W/

Limit of a function along a curve

• f(x,y) = x+y
• r(t) = <3t,t>

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Surfaces (review)

• EQ: $3x+2y+z=4$,
  • $x^2+y^2+{\left(\frac{z}{2}\right)}^2=1$
• Func: $f(x,y)=4-3x-2y$,
  • $z=4-3x+2y$
• Param: $r(s,t)=<x(s,t),y(s,t),z(s,t)>$

Limits of func in 2d

• In 1d 2 dir, limit from the left and right
• 2d? Lim of @param curve along r(t) @ P

Eg) Limit of func along a curve @ a pt?

Given:

• $f(x,y)=1+x+2y$)
• $P=(1,1)$, $r(t)=t(1,1)+(1,1)$
• Where r(t) some @param cruve, only restrict is need 2 pass 1,1-
  • What is limit of f along r @ P?

1. What t cord is (1,1), $t_0=0$
2. ${\lim }_{t\to 0}f(r(t))=f(t+1,t+1)$
   1. Or ${\lim }_{t\to 0}(1+(t+1)+2(t+1))={\lim }_{t\to 0}(4+3t)=4$

eg.2

• $f(x,y)=\frac{xy}{x^2+y^2}$
• P=(0,0)

So:

• r(t) = t<2,0> = <2t,0>
  • Hence $t_0=0$
• Get ${\lim }_{t\to 0}f(r(t))$, using the r(t) values of $\hat{i},\hat{j}$
• say rebind: $r(t)=t<3,4>$ or $<3t,4t>$, again apply ${\lim }_{t\to 0}f(r(t))=\frac{12}{25}$
• This tells us how we approach some cliff from a certain direction (where each direciton can have a different limit)

limit laws{cal3}

Example:

• $f(x)=e^x,g(x,y)=\frac{2x^2+y}{x+y}$
• Then $f(g(x,y))$? Simplfy plug in $e^{g(x,y)}$
• What about $g(f(x),y)$? Then js plug in $e^x$ for x and keep y

Conic secitons!

• Spheres
• Ellipces
• hypeberla
• parabolas