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# jacobianhandout - Jacobian Transformation Shubhodeep...

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Jacobian Transformation Shubhodeep Mukherji May 22, 2011 1 Basics of Multi-Cal 1.1 Partial Derivative Given a function f , when taking the partial derivative with respect to one variable, treat all the other variables as constants. Example 1 Find ∂f ∂x and ∂f ∂y for the function f ( x, y ) = x 2 + 3 xy + y - 1 . 1.2 Fubini’s Theorem If f ( x, y ) is continuous throughout rectangular region R : a x b, c y d , then ZZ R f ( x, y ) dA = Z d c Z b a f ( x, y ) dxdy = Z b a Z d c f ( x, y ) dydx. 1.3 Finding Limits of Integration If calculating RR R f ( x, y ) dA first wrt y and then wrt x , do the following: 1. Sketch region of integration and label bounding curves 2. Find y-limits of integration 3. Find x-limits of integration Example 2 Calculate the area of region R bounded by the curves x 2 + y 2 = 1 and x + y = 1 . 1

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2 Rotation 2.1 Mass and First Moment Formulas Mass: M = ZZ R δ ( x, y ) dA First Moments: M x = ZZ R ( x, y ) dA, M y = ZZ R ( x, y ) dA Center of Mass: x = M y M , y = M x M 2.2 Second Moment Formulas Second moment is rotational inertia.
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