MATH115 GEOMETRY AND CALCULUS

Workshop Exercises 3

• What is the greatest value of $4x-2y+3z$ on the sphere $x^2+y^2+z^2=2$?

• Find the equations of the normal line and the tangent plane to the surface $z^2=$ $x^2+2y^2+19$ at the point $(3,2,6)$.

• Is the following vector-valued function expressible as $\nabla \varphi$ for some $\varphi$? If so, find it.

  $$𝒇(x,y,z)=(2y-z,\mathrm{\hspace{0.33em}3}z+2x,-x+3y).$$

• Let $u=3x-2y$, $v=3y-4x$. Show that

  $$\frac{{\partial }^{2}f}{\partial x\partial y}=-6\frac{{\partial }^{2}f}{\partial u^2}+17\frac{{\partial }^{2}f}{\partial u\partial v}-12\frac{{\partial }^{2}f}{\partial v^2}$$

  and

  $$\frac{{\partial }^{2}f}{\partial u\partial v}=6\frac{{\partial }^{2}f}{\partial x^2}+17\frac{{\partial }^{2}f}{\partial x\partial y}+12\frac{{\partial }^{2}f}{\partial y^2}$$

• Let $\rho (𝒓)=|𝒓-{𝒓}_0|$, where $𝒓=(x,y,z)$ and ${𝒓}_0=(x_0,y_0,z_0)$, with $x_0,y_0,z_0$ fixed. Show that $\nabla \rho =\frac{1}{\rho (𝒓)}(𝒓-{𝒓}_0)$. (Write $\rho$ in terms of $x,y,z$.)