Sec2Sess35

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Unformatted notes below. Notes on my shorthand method

Gradient, definition and perpendicular to the level surface

Recall Chain Rule

w = w(x,y,z)

x=x(t)

y=y(t)

z=z(t)

dw/dt=wx*dx/dt+wy*dy/dt+wz*dz/dt

dw/dt = g(w) . d(->_r)/d(t) where g() is the gradient, "." is the dot product, and ->_x is the velocity vector g(w) = <wx,wy,wz>

GRADIENT OF W at some point (x,y,z)

later this will be termed the "vector feild"

d(->_r)/d(t) = <d(x)/d(t),d(y)/d(t),d(z)/d(t)>

Theorem: g(w) is perpendicular to the level surface {w = constant} It is perpendicular to the level curves in a function such as f(x,y) = c

Example 1:

linear function

w = a1*x+a2*y+a3*z

g(w) = <a1,a2,a3>

a1 = pd(w)/pd(x)

a2 ... etc.

level surface

a1*x+a2*y+a3*z=c , the surface is in the form of a plane plane with normal vector that is equal to the gradient <a1,a2,a3>

Example 2:

w = x^2+y^2

w=c is a circle when graphed in 2d. center is at 0,0

x^2+y^2=c

g(w) = <2x,2y>

gradient is perpendicular to points along the circumference of the circle