Integral of a function of various variables

[Jhenrique]

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

 

[Mark44]

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

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).