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jacobian - Jacobian Transformation Shubhodeep Mukherji...

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Jacobian Transforma- tion Shubhodeep Mukherji Basics of Multi-Cal Rotation Ellipse Problem Transformation Jacobian in Action Jacobian Transformation Shubhodeep Mukherji Multivariate Calculus Independent Study Science Academy of South Texas May 22, 2011
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Jacobian Transforma- tion Shubhodeep Mukherji Basics of Multi-Cal Rotation Ellipse Problem Transformation Jacobian in Action Partial Derivatives Math!
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Jacobian Transforma- tion Shubhodeep Mukherji Basics of Multi-Cal Rotation Ellipse Problem Transformation Jacobian in Action Partial Derivatives Math! Use the symbol to distinguish partial derivative
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Jacobian Transforma- tion Shubhodeep Mukherji Basics of Multi-Cal Rotation Ellipse Problem Transformation Jacobian in Action Partial Derivatives Math! Use the symbol to distinguish partial derivative Given a function f , when taking the partial derivative with respect to one variable, treat all the other variables as constants.
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Jacobian Transforma- tion Shubhodeep Mukherji Basics of Multi-Cal Rotation Ellipse Problem Transformation Jacobian in Action Partial Derivatives Math! Use the symbol to distinguish partial derivative Given a function f , when taking the partial derivative with respect to one variable, treat all the other variables as constants. Example Find f x and f y for the function f ( x , y ) = x 2 + 3 xy + y - 1.
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Jacobian Transforma- tion Shubhodeep Mukherji Basics of Multi-Cal Rotation Ellipse Problem Transformation Jacobian in Action Multiple Integrals Math! Same principle as for Partial Derivatives Example Z 3 0 Z 2 0 ( 4 x - y 2 ) dydx
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Jacobian Transforma- tion Shubhodeep Mukherji Basics of Multi-Cal Rotation Ellipse Problem Transformation Jacobian in Action Area and Fubini’s Theorem Math! Area Area of a closed, bounded region R is A = ZZ R dA . Note: dA is called the ”‘area element”’
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Jacobian Transforma- tion Shubhodeep Mukherji Basics of Multi-Cal Rotation Ellipse Problem Transformation Jacobian in Action Area and Fubini’s Theorem Math! Area Area of a closed, bounded region R is A = ZZ R dA . Note: dA is called the ”‘area element”’ Fubini’s Theorem If f ( x , y ) is continuous throughout rectangular region R : a x b , c y d , then
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Jacobian Transforma- tion Shubhodeep Mukherji Basics of Multi-Cal Rotation Ellipse Problem Transformation Jacobian in Action Area and Fubini’s Theorem Math! Area Area of a closed, bounded region R is A = ZZ R dA . Note: dA is called the ”‘area element”’ 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 .
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Jacobian Transforma- tion Shubhodeep Mukherji Basics of Multi-Cal Rotation Ellipse Problem Transformation Jacobian in Action Finding Limits of Integration Math!
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