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Tangent Planes and Normal Vectors

washam13
washam13
Tangent Planes and Normal Vectors
Apr 6 2011, 7:19 PM EDT | Post edited: Apr 6 2011, 7:23 PM EDT
tangent line to a curve on a surface z = f(x,y)
L(x,y) = f(x0,y0) + fx(x0,y0)(x-xo) + fy(x0,y0)(y-y0)

tangent plane
z + f(x0,y0) + fx(x0,y0)(x-x0) + fy(xo0,yo)(y-y0)

normal line
x= x0 + fx(x0,y0)
y= y0 + fy(x0,y0)
z= f(x0,y0)

tangent planes to level surfaces
C= F(x0,y0,z0)
Fx(x0,y0,z0)(x-x0) + Fy(x0,y0,z0)(y-y0) + Fz(x0,y0,z0)(z-z0) = 0
gradient F(x0,y0,z0) = n-- =Fxi + Fyj + Fzk

tangent line to a curve of intersection of 2 surfaces
gradient F(x,y,z) X gradient G(x,y,z) = tangent vector
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