Sec2Sess36

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Unformatted notes below. Notes on my shorthand method

Proof of Gradient Perpendicular to Level Curves and Surfaces Note from class: Gradient always points towards higher values of a function w. Therefore it is perpendicular to tangent line of function.

Take a curve r-> = r->(t) that stays on the level surface

w(x,y,z)=c

velocity vector is tangent to curve of function

w = c for level surface

velocity v-> d(r->)/d(t) is tangent to the level w=c.

By chain rule: dw/dt = g(w) . d(r->)/d(t)

= g(w) . v-> = 0 since w(t) = c = constant

dot product of 0 prooves that vectors are perpendicular

So g(w) is perpendicular to v->

This is true for any motion on w=c. Therefore v-> can be any vector tangent to w=c and g(w) is still perpendicular.

tangent plane to level plane

given any vector v-> tangent to level plane, g(w) is perpendicular to the tangent plane at that point.

so g(w) is perpendicular to the tangent plane to the level