exam1formulas

exam1formulas - Formula sheet for exam 1 in calc IV, summer...

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Formula sheet for exam 1 in calc IV, summer 2007 Polar coordinates x = r cos θ y = r sin θ r 2 = x 2 + y 2 dA = rdrdθ Cylindrical coordinates x = r cos θ y = r sin θ r 2 = x 2 + y 2 dV = rdzdrdθ Spherical coordinates x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ ρ 2 = x 2 + y 2 + z 2 dV = ρ 2 sin φdρdθdφ Change of variables in 2 dimensions RR R f ( x,y ) dA x,y = RR S ( f T )( u,v ) | J ( T ) | dA u,v ; J ( T ) = ( x,y ) ( u,v ) = ± ± ± ± ∂x ∂u ∂x ∂v ∂y ∂u ∂y ∂v ± ± ± ± , the Jacobian. Change of variables in 3 dimensions RRR R f ( x,y,z ) dV x,y,z = RRR S ( f T )( u,v,w ) | J ( T ) | dV u,v,w ; J ( T ) = ( x,y,z ) ( u,v,w ) = ± ± ± ± ± ± ∂x ∂u ∂x ∂v ∂x ∂w ∂y ∂u ∂y ∂v ∂y ∂w ∂z ∂u ∂z ∂v ∂z ∂w ± ± ± ± ± ± Total mass of a mass distribution (density) ρ ( x,y ) over a region R of R 2 is m = RR R ρ ( x,y ) dA. The moment about the x -axis is M x = RR R ( x,y ) dA ; the moment about the y - axis is M y = RR R ( x,y ) dA . The center of gravity is ² M y m , M x m ³ . Total mass of a mass distribution (density) ρ ( x,y,z ) over a region R of R 3 is m = RRR R ρ ( x,y,z ) dV. The moment of inertia about an axis (line) l is RRR R dist( l, ( x,y,z )) 2 ρ ( x,y,z ) dV If f ( x,y ) is a joint probability density function for random variables
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This note was uploaded on 10/18/2011 for the course MATH S1202Q taught by Professor Peters during the Spring '11 term at Columbia.

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