Sec2Sess35
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