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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 R f ( x, y ) dA x,y = 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 R f ( x, y, z ) dV x,y,z = 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 = R ρ ( x, y ) dA. The moment about the x -axis is M x = R ( x, y ) dA ; the moment about the y - axis is M y = R ( x, y ) dA . The center of gravity is M y m , M x m . Total mass of a mass distribution (density) ρ ( x, y, z ) over a region R of R 3 is m = R ρ ( x, y, z ) dV. The moment of inertia about an axis (line) l is 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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