FactSheet_midterm1_Math20E(2)

FactSheet_midterm1_Math20E(2) - The tangent plane at ( x ,y...

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Fact Sheet for Midterm 1 Vector parametrization of the line through ( x 0 ,y 0 ,z 0 ) in the direction v = ( a,b,c ): r ( t ) = ( x 0 ,y 0 ,z 0 ) + t ( a,b,c ). The dot product of v = ( a 1 ,b 1 ,c 1 ) and w = ( a 2 ,b 2 ,c 2 ): v · w = a 1 ,a 2 + b 1 b 2 + c 1 c 2 . v · w = || v |||| w || cos θ , with θ the angle between v and w . The orthogonal projection of u on v is proj v ( u ) = u · v v · v v . The cross product of v = ( a 1 ,b 1 ,c 1 ) and w = ( a 2 ,b 2 ,c 2 ): v × w = ± ± ± ± ± ± i j k a 1 b 1 c 1 a 2 b 2 c 2 ± ± ± ± ± ± = ± ± ± ± b 1 c 1 b 2 c 2 ± ± ± ± i - ± ± ± ± a 1 c 1 a 2 c 2 ± ± ± ± j + ± ± ± ± a 1 b 1 a 2 b 2 ± ± ± ± k The area of the parallelogram spanned by v and w is || v × w || . The volume of the parallelepiped spanned by u , v and w is | u · ( v × w ) | . An equation of the plane through ( x 0 ,y 0 ,z 0 ) with normal vector n = ( a,b,c ) can be written in vector form as n · ( x - x 0 ,y - y 0 ,z - z 0 ) = 0; in scalar form as ax + by + cz = ax 0 + bx 0 + cz 0 . Let f be a function from R n to R , then the gradient of f is f = ( ∂f ∂x 1 , ··· , ∂f ∂x n ). Let f be a C 1 function from R n to R m , then the derivative of f is the m × n -matrix D f = ∂f 1 ∂x 1 ··· ∂f 1 ∂x n . . . . . . ∂f m ∂x 1 ··· ∂f m ∂x n
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Unformatted text preview: The tangent plane at ( x ,y ,f ( x ,y )) to the surface z = f ( x,y ) is given by z = f ( x ,y ) + ( f x ( x ,y ) ) ( x-x ) + f y ( x ,y ) ( y-y ). For g : R n R m , f : R m R p , g ( x ) = y , and h ( x ) = f ( g ( x )), D h ( x ) = [ D f ( y )] [ D g ( x )] . The directional derivative of f : R 3 R at x in the direction of v is given by D f ( x ) v = f ( x ) v . The tangent plane to the level surface f ( x,y,z ) = k at ( x ,y ,z ) has equation f ( x ,y ,z ) ( x-x ,y-y ,z-z ) = 0, provided the gradient is not the zero vector....
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