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:

https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-b-chain-rule-gradient-and-directional-derivatives/session-37-example/MIT18_02SC_we_17_comb.pdf

https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-b-chain-rule-gradient-and-directional-derivatives/session-37-example/MIT18_02SC_pb_44_quest.pdf