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Unformatted text preview: Jacobian Transforma tion Shubhodeep Mukherji Basics of MultiCal Rotation Ellipse Problem Transformation Jacobian in Action Jacobian Transformation Shubhodeep Mukherji Multivariate Calculus Independent Study Science Academy of South Texas May 22, 2011 Jacobian Transforma tion Shubhodeep Mukherji Basics of MultiCal Rotation Ellipse Problem Transformation Jacobian in Action Partial Derivatives Math! Jacobian Transforma tion Shubhodeep Mukherji Basics of MultiCal Rotation Ellipse Problem Transformation Jacobian in Action Partial Derivatives Math! Use the symbol ∂ to distinguish partial derivative Jacobian Transforma tion Shubhodeep Mukherji Basics of MultiCal 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. Jacobian Transforma tion Shubhodeep Mukherji Basics of MultiCal 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. Jacobian Transforma tion Shubhodeep Mukherji Basics of MultiCal Rotation Ellipse Problem Transformation Jacobian in Action Multiple Integrals Math! Same principle as for Partial Derivatives Example Z 3 Z 2 ( 4 x y 2 ) dydx Jacobian Transforma tion Shubhodeep Mukherji Basics of MultiCal 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”’ Jacobian Transforma tion Shubhodeep Mukherji Basics of MultiCal 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 Jacobian Transforma tion Shubhodeep Mukherji Basics of MultiCal 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 . Jacobian Transforma tion Shubhodeep Mukherji Basics of MultiCal Rotation Ellipse Problem Transformation Jacobian in Action Finding Limits of Integration Math!...
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This note was uploaded on 08/03/2011 for the course ECON 101 taught by Professor Profeessor during the Spring '11 term at Aachen University of Applied Sciences.
 Spring '11
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