User:Selfworm/Math3

Take for example,


 * $$u(x, y) = x^6 - 15{x^4}{y^2} + 15{x^2}{y^4} - y^6\,$$

so,


 * $$u_x(x, y) = 6x^5 - 60{x^3}{y^2} + 30x{y^4} - 0\,$$
 * $$u_y(x, y) = 0 - 30yx^4 + 60{x^2}{y^3} - 6y^5\,$$

and using the $$y = 0\,$$ substitution so as to limit the domain of the example to the real numbers yields,


 * $$u_x(x, y) = 6x^5 - 60{x^3}{0^2} + 30x{0^4} - 0\,$$
 * $$u_y(x, y) = 0 - 30(0)x^4 + 60{x^2}{(0)^3} - 6(0)^5\,$$

and finally the main result can be written as,


 * {| style="background:none;"


 * $$f(z)\,$$
 * $$= \int (6z^5) - i(0)\,dz\,$$
 * $$= z^6\,$$
 * }
 * $$= z^6\,$$
 * }
 * }

Note that the substitution $$y = 0\,$$ makes the cancellation of terms easy and reduces the possibility of errors and that we did not need to know what $$v(x, y)\,$$ was in order to derived $$f(z)\,$$.