Duf = d/ds[f(x0 + su1 , y0 + su2, z0 + su3)] domain
Duf = fx(x0, y0, z0)u1 + fy(x0, y0, z0)u2 + fz(x0, y0, z0)u3
Duf = fx(x0, y0)cos(theta) + fy(x0, y0)sin(theta)

gradients
gradients are vectors
gradient f = fx i + fy j + fz k (f(x,y,z)) , gradient f = fx i + fy j - k when (z= f(x,y))

Duf = gradient f * u u is the unit vector
Duf in the direction of gradient f at P has the largest value and equals ll gradient f ll at P...... the opposite direction is the lowest value

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differentiable if fx(x0, y0) and fy(x0, y0) exist and


lim [delta f - fx(x0,y0)(delta x) - fy(x0,y0)(delta y)] / sqrt( (delta x)^2 + (delta y)^2 ) = 0

delta x --> 0 delta y --> 0
if differentiable, then continuous

Differentials:
dz = fx(x0,y0)dx + fy(x0,y0)dy l
dW = fx(x0,y0,z0)dx + fy(x0,y0,z0)dy + fz(x0,y0,z0) l total differential of F
delta f is approximated to df
delta z is approximated to dz

Local linear approximation
L(x,y) = f(x0,y0) + fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0)

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