Solve df=partialT/partialxdx+partialf/partialydy

$$d f = \frac { \partial T } { \partial x } d x + \frac { \partial f } { \partial y } d y$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&f=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \frac{\mathrm{d}(T)}{\mathrm{d}x}dx from both sides.

$$df-\frac{\mathrm{d}(T)}{\mathrm{d}x}dx=\frac{\mathrm{d}(f)}{\mathrm{d}y}dy$$

Subtract \frac{\mathrm{d}(f)}{\mathrm{d}y}dy from both sides.

$$df-\frac{\mathrm{d}(T)}{\mathrm{d}x}dx-\frac{\mathrm{d}(f)}{\mathrm{d}y}dy=0$$

Reorder the terms.

$$-dx\frac{\mathrm{d}(T)}{\mathrm{d}x}-dy\frac{\mathrm{d}(f)}{\mathrm{d}y}+df=0$$

Combine all terms containing d.

$$\left(-x\frac{\mathrm{d}(T)}{\mathrm{d}x}-y\frac{\mathrm{d}(f)}{\mathrm{d}y}+f\right)d=0$$

The equation is in standard form.

$$fd=0$$

Divide 0 by f.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&f=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \frac{\mathrm{d}(T)}{\mathrm{d}x}dx from both sides.

$$df-\frac{\mathrm{d}(T)}{\mathrm{d}x}dx=\frac{\mathrm{d}(f)}{\mathrm{d}y}dy$$

Subtract \frac{\mathrm{d}(f)}{\mathrm{d}y}dy from both sides.

$$df-\frac{\mathrm{d}(T)}{\mathrm{d}x}dx-\frac{\mathrm{d}(f)}{\mathrm{d}y}dy=0$$

Reorder the terms.

$$-dx\frac{\mathrm{d}(T)}{\mathrm{d}x}-dy\frac{\mathrm{d}(f)}{\mathrm{d}y}+df=0$$

Combine all terms containing d.

$$\left(-x\frac{\mathrm{d}(T)}{\mathrm{d}x}-y\frac{\mathrm{d}(f)}{\mathrm{d}y}+f\right)d=0$$

The equation is in standard form.

$$fd=0$$

Divide 0 by f.

$$d=0$$

Solve for T (complex solution)

$$T\in \mathrm{C}$$

$$f=0\text{ or }d=0$$

Solve for T

$$T\in \mathrm{R}$$

$$f=0\text{ or }d=0$$