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Integral of a function of various variables

  1. Mar 6, 2014 #1
    When I have a function z = f(x, y), i. e., a function of various variables, the differential form of z is: [tex]dz = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy[/tex] or the derivative of z is: [tex]\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}[/tex] So, analogously, if I take the integral of z wrt t, so how come to be the expression for f?
  2. jcsd
  3. Mar 6, 2014 #2


    Staff: Mentor

    When you take the derivative of z with respect to t, you are tacitly assuming that both x and y are functions of t alone. That is, that x = x(t) and y = y(t).

    With that assumption, ##\int f(x, y) dt = \int f(x(t), y(t)dt##. As far as I know, that can't be simplified without knowing more about x(t) and y(t).
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