The first example we see below is the graph of z = sin(x) sin(y) It's a function of x and y You can use the following applet to explore 3D graphs and even create your own, using variables x and y You can also toggle between 3D Grapher mode and Contour mode Things to do 1 Choose any of the preset 3D graphs using the drop down box at the top 2 You can enterLevel surfaces For a function $w=f(x,\,y,\,z) \, U \,\subseteq\, {\mathbb R}^3 \to {\mathbb R}$ the level surface of value $c$ is the surface $S$ in $U \subseteqExercise 1 Verify that both u= log(x2y2) and u= arctan(y=x) are solutions of Laplace's equation u xx u yy= 0 If u= log(x2 y2), then by the chain rule u x= 2x x 2 y) u xx= (x2 y2)(2) (2x)(2x) (x 2 y) = 2y2 2x2 (x y2)2;
Solutions To Implicit Differentiation Problems
