14'7 Substitutions in Multiple Integrals

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14.7 Substitutions in Multiple Integrals
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Definition The Jacobian

• Suppose that a region G in the uv-plane is
  transformed one-to-one into the region R in the
  xy-plane by the equations
• xg(u,v) and yh(u,v)
• The Jacobian is defined by

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The Integrals

• xg(u,v) and yh(u,v)

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Definition The Jacobian

• xg(u,v,w) yh(u,v,w) zk(u,v,w)
• The Jacobian is defined by

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The Integrals

• xg(u,v,w) yh(u,v,w) zk(u,v,w)