The Unapologetic Mathematician

Mathematics for the interested outsider

The Inverse Function Theorem

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November 18, 2009 - Posted by | Analysis, Calculus

4 Comments »

  1. […] The Implicit Function Theorem II Okay, today we’re going to prove the implicit function theorem. We’re going to think of our function as taking an -dimensional vector and a -dimensional vector and giving back an -dimensional vector . In essence, what we want to do is see how this output vector must change as we change , and then undo that by making a corresponding change in . And to do that, we need to know how changing the output changes , at least in a neighborhood of . That is, we’ve got to invert a function, and we’ll need to use the inverse function theorem. […]

    Pingback by The Implicit Function Theorem II « The Unapologetic Mathematician | November 20, 2009 | Reply

  2. […] all of these cases, we know that the inverse function exists because of the inverse function theorem. Here the Jacobian determinant is simply the derivative , which we’re assuming is everywhere […]

    Pingback by Change of Variables in Multiple Integrals I « The Unapologetic Mathematician | January 5, 2010 | Reply

  3. […] assume that is injective and that the Jacobian determinant is everywhere nonzero on . The inverse function theorem tells us that we can define a continuously differentiable inverse on all of the image […]

    Pingback by Change of Variables in Multiple Integrals II « The Unapologetic Mathematician | January 6, 2010 | Reply

  4. […] the inverse function theorem from multivariable calculus: if is a map defined on an open region , and if the Jacobian of has […]

    Pingback by The Inverse Function Theorem « The Unapologetic Mathematician | April 14, 2011 | Reply


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