# F xy is normal to level curve fxy k f xyz is normal

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f (x,y) is normal to level curve f(x,y) = k; f (x,y,z) is normal to level surface f(x,y,z) = k. D u f (x,y,z) = f (x,y,z) . ( u / || u ||) = f (x,y,z) . û 3. f(x,y) is differentiable at the point x 0 ,y 0 if the limit as ∆x, ∆y tends to 0, 0 of [ {f (x 0 +∆x, y 0 +∆y) - f(x 0 , y 0 )} - { f (x 0 , y 0 ) . ∆x, ∆y } ] / || ∆x, ∆y || = 0 . 4. Differentials: dw = ( w/ x).dx + ( w/ y).dy + ( w/ z).dz . ∆w ( w/ x). ∆x + ( w/ y).∆y + ( w/ z).∆z . f (x 0 +∆x, y 0 +∆y) f(x 0 , y 0 ) + f x (x 0 , y 0 ).∆x + f y (x 0 , y 0 ).∆y 5. Chain rules: (dw/dt) = ( w/ x). (dx/dt) + ( w/ y). (dy/dt) + ( w/ z) (dz/dt). ( w/ u) = ( w/ x). ( x/ u) + ( w/ y). ( y/ u) + ( w/ z) ( z/ u). ( w/ v) = ( w/ x). ( x/ v) + ( w/ y). ( y/ v) + ( w/ z) ( z/ v). 6. Normal to the surface z = f(x,y) at x 0 , y 0, f(x 0 ,y 0 ) is -f x (x 0 , y 0 ) , -f y (x 0 , y 0 ) , 1 . Normal to the surface r (u,v) = x(u,v), y(u,v), z(u,v) is ( r / u)×( r / v). 7. At a gradient–zero point x 0 , y 0 we have f (x 0 , y 0 ) = 0 , i.e., f/ x = 0 & f/ y = 0. If x 0 , y 0 is a gradient-zero point and D = {f xx (x 0 , y 0 ). f xx (x 0 , y 0 )} - {f xy (x 0 , y 0 )} 2 , (a) D > 0 and f xx (x 0 , y 0 ) > 0 will tell us we have a local minimum at x 0 , y 0 , (b) D > 0 and f xx (x 0 , y 0 ) < 0 will tell us we have a local maximum at x 0 , y 0 , (c) D < 0 will tell us we have a local saddle point at x 0 , y 0 , and (d) D = 0 will tell us nothing because the point x 0 , y 0 could be any of the above. 8. x 0 , y 0 is a critical point of f(x,y) if f (x 0 ,y 0 ) = 0 or if f (x,y) does not exist at x 0 ,y 0 . The global maximum and global minimum of f(x,y) will be achieved at a critical point or at a point on the boundary of the domain of f(x,y).
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