Sec2Sess37
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Unformatted notes below. Notes on my shorthand method
Example: find the tangent plane to surface x^2+y^2-z^2 = 4 at point (2,1,1)?
Level set w=4, where w=x^2+y^2-z^2.
Note: "level set" is result of the tangent plane to surface equation.
gradient: g(w) = <2x, 2y, -2z>
normal vector = <4,2,-2> when point (2,1,1) is applied
so eq. is 4x+2y-2z=8 because when point (2,1,1) is applied
to 4x+2y-2z than the result is 8.
Another way:
dw= 2xdx+2ydy-2zdz
= 4dx + 2dy - 2dz (at (2,1,1))
change_w ~= 4*change_x+2*change_y-2*change_z
replace function by tangent approximation
So: level: change_w = 0
its tangent plane: 4*change_x+2*change_y-2*change_z=0
= 4(x-2)+2(y-1)-2(z-1)=0
Example problems: