a. Calculate (∂f/∂x)y, (∂f/∂y)x, (∂²f/∂x²)y, (∂²f/∂y²)x, (∂(∂f/∂x)y/∂y)x, and (∂(∂f/∂y)x/∂x)y for the function f(x, y) = ye^x + xln y. b. Determine if f(x, y) is a state function of the variables x and y. c. If f(x, y) is a state function of the variables x and y, what is the total differential df?

Question

a. Calculate

[latex]\left( \frac{\partial f}{\partial x} \right)_{y},\left( \frac{\partial f}{\partial y} \right)_{x},\left( \frac{\partial^{2} f}{\partial x^{2}} \right)_{y},\left( \frac{\partial^{2} f}{\partial y^{2}} \right)_{x},\left( \frac{\partial\left( \frac{\partial f}{\partial x} \right)_{y}}{\partial y} \right)_{x}, and \left(\frac{\partial\left( \frac{\partial f}{\partial y} \right)_{x}}{\partial x} \right)_{y}[/latex]

for the function [latex] f\left( x,y \right)=ye^{x}+x y+x\ln y[/latex].

b. Determine if [latex]f\left( x,y \right)[/latex] is a state function of the variables x and y.

c. If [latex] f\left( x,y \right)[/latex] is a state function of the variables x and y , what is the total differential df?

Accepted Answer

a. [latex]\left( \frac{\partial f}{\partial x} \right)_{y}=ye^{x} + y + \ln y, \left( \frac{\partial f}{\partial y} \right)_{x}=e^{x} +x+\frac{x}{y}[/latex]

[latex]\left( \frac{\partial^{2}f }{\partial x^{2}} \right)_{y}= ye^{x}, \left( \frac{\partial^{2}f }{\partial y^{2}} \right)_{x}=-\frac{x}{y^{2}}[/latex]

[latex]\left( \frac{\partial \left( \frac{\partial f}{\partial x} \right)_{y}}{\partial y} \right)_x=e^{x} + 1+ \frac{1}{y}, \left( \frac{\partial \left( \frac{\partial f}{\partial y} \right)_{x}}{\partial x} \right)_{y} = e^{x} + 1+\frac{1}{y} [/latex]

b. Because we have shown that

[latex]\left( \frac{\partial \left( \frac{\partial f }{\partial x} \right)_{y}}{\partial y} \right)_{x} = \left( \frac{\partial \left( \frac{\partial f}{\partial y} \right)}{\partial x} \right)_{y} [/latex]

[latex] f\left( x,y \right)[/latex] is a state function of the variables x and y. Generalizing this result, any well-behaved function that can be expressed in analytical form is a state function.
c. The total differential is given by

[latex]df = \left( \frac{\partial f}{\partial x} \right)_{y} dx + \left( \frac{\partial f}{\partial y} \right)_{x} dy [/latex]

[latex]=\left( ye^{x} + y +\ln y \right) dx+ \left( e^{x} +x+\frac{x}{y}\right)dy[/latex]

Question 3.1: a. Calculate (∂f/∂x)y, (∂f/∂y)x, (∂²f/∂x²)y, (∂²f/∂y²)x, (∂(...

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