calculus1 - The following is a list of formulae and...

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Unformatted text preview: The following is a list of formulae and theorems of math 23 to help you to memorize the materials and thus no proofs or examples are included. If a formula holds for both 2D and 3D cases and the expressions are similar, we only state the 3D case. (1) Distance formula: given two points P = ( x 1 ,x 2 ,x 3 ) ,Q = ( y 1 ,y 2 ,y 3 ) ∈ R 3 , then the Euclidean distance of p and Q is d ( P,Q ) = | PQ | = p ( x 1- y 1 ) 2 + ( x 2- y 2 ) 2 + ( x 3- y 3 ) 2 . (2) Equation of spheres and description of balls: The equation of the sphere with center ( a,b,c ) and radius r is ( x- a ) 2 + ( y- b ) 2 + ( z- c ) 2 = r 2 or x 2 + y 2 + z 2- 2 ax- 2 by- 2 cz + a 2 + b 2 + c 2- r 2 = 0 . Please see the later classification of quadric surfaces. The (closed) ball with center P = ( a,b,c ) and radius r is B ( P,r ) = B r ( P ) = { ( x,y,z ) ∈ R 3 | ( x- a ) 2 + ( y- b ) 2 + ( z- c ) 2 ≤ r 2 } . (3) Properties of vectors: given vectors v = h x,y,z i and w = h a,b,c i we have • The length of v is | v | = p x 2 + y 2 + z 2 . • v ± w = h x ± a,y ± b,z ± c i . • If α is a scalar, then αv = h αx,αy,αz i . • v is the zero vector if and only if x = y = z = 0. • If v 6 = 0, then the unit vector in the direction of v is 1 | v | v . • The standard basis vectors i , j , k have components i = h 1 , , i , j = h , 1 , i , k = h , , 1 i . We can write v as a linear combination v = x i + y j + z k . (4) Products of vectors: given vectors v = h x,y,z i and w = h a,b,c i , the dot product of v and w is the scalar v · w = xa + yb + zc. Properties of dot product: • | v | 2 = v · v . • v · w = w · v . • For scalars α,β and vectors v 1 ,v 2 ,w we have ( αv 1 + βv 2 ) · w = αv 1 · w + βv 2 · w. • Let θ be the angle between v and w . Then v · w = | v || w | cos θ. • For nonzero vectors v and w , they are perpendicular if and only if v · w = 0. • The scalar projection of w onto v is comp v w = v · w | v | and the vector of w onto v is proj v w = v · w | v | v | v | = v · w | v | 2 v. 1 The cross product of two 3D vectors v = h x,y,z i and w = h a,b,c i is the vector v × w = det i , j , k x, y, z a, b, c = ( yc- bz ) i + ( za- xc ) j + ( xb- za ) k . Properties of dot product: • v × w =- w × v . • For scalars α,β and vectors v 1 ,v 2 ,w we have ( αv 1 + βv 2 ) × w = αv 1 × w + βv 2 × w. • Let θ be the angle between v and w . Then | v × w | = | v || w | sin θ. The vector v × w is perpendicular to both v and w and the triple ( v,w,v × w ) satisfy the right hand rule. Furthermore, | v × w | is the area of the parallelo- gram spanned by v and w . • For nonzero vectors v and w , they are parallel if and only if v × w = 0....
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This note was uploaded on 02/09/2008 for the course MATH 23 taught by Professor Yukich during the Spring '06 term at Lehigh University .

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calculus1 - The following is a list of formulae and...

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