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Math Help - Test for exact equations / how to interpret definition

  1. #1
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    Test for exact equations / how to interpret definition

    The following two definitions are given in my lecture notes:

    First.

    Let u : R x R --> R be a function with continuous partial derivatives and let y: R--->R be differentiable

    What does u: R x R ---> R mean?
    and what does y: R--->R mean

    secondly, to test if a function is exact, we have:

    If M and N are continuous and have continuous first order partial derivatives in some rectangular domain then the differential form Mdx + Ndy is exact in R
    if and only if
    m/y = n/∂x in R

    I understand everything up until the "Rectangular domain". What is that supposed to tell me? How do I know if a domain is rectangular?

    Thanks.
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  2. #2
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    Re: Test for exact equations / how to interpret definition

    Hey 99.95.

    When you see u: R x R ----> R it means that u is a function taking a point in R x R (which is R^2 or a 2D (x,y) point) and then producing a real number in R.

    An example is u(x,y) = x + y. We take two real numbers x and y and produce a real number u(x,y).

    Whenever you see A x B, it means that you form all pairs of points in a with points in b and your new point will be (a,b). You continue to do this if you have more cross products like say A x B x C or A x B x C x D and so on.

    A rectangular domain is just some rectangular region. If the domain is 2D (in your case it is), then its just some rectangle that has its boundary aligned with the x and y axes.
    Thanks from 99.95
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  3. #3
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    Re: Test for exact equations / how to interpret definition

    thank you, that makes perfect sense.
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