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Partial derivatives

  1. Jan 7, 2010 #1
    1. The problem statement, all variables and given/known data
    A mapping f from an open subset S of Rn into Rm is called smooth if it has continuous partial derivatives of all orders. However, when the domain S is not open one cannot usually speak of partial derivatives. Why?

    2. Relevant equations

    3. The attempt at a solution
    In the 1 dimensional case there are not partial derivatives and we can consider the derivative of a function on a closed set by just using the derivative from the left if we are at the left boundary point of the interval. In 2 dimensions I tried creating a counter example, but no luck yet. In the definition of the partial derivative we already assume the domain to be open.
    Last edited: Jan 7, 2010
  2. jcsd
  3. Jan 7, 2010 #2
    Can you come up with a natural well-defined analog of a one-sided limit in n-dimensions?
  4. Jan 7, 2010 #3
    In n-dimensions the analog of the derivative is the total derivative (i.e., gradient). When we consider the partial derivative or directional derivative in the direction of a unit coordinate vector we use a similar definition to that of the 1 dimensional derivative. In the 1 sided limit in 1 dimension we can approach the a point from 1 direction only. What I have in my mind is that we can approach a point in n-dimensions, for example, along the right hand side of the line that passes through the point where the partial derivative is taken in the direction of the coordinate unit vector.
  5. Jan 7, 2010 #4
    Surely you've seen an example of a function of the plane such that the limits along the x and y axis exist and are different. Which one would you choose?
  6. Jan 7, 2010 #5
    yes, f(x,y)=(x+y)/(x-y).
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