Exercises

1.
Evaluate the directional derivative of f$\left(\vphantom{ x,y,z}\right.$x, y, z$\left.\vphantom{ x,y,z}\right)$ = 3x - 5y + 2z at $\left(\vphantom{ 2,2,1}\right.$2, 2, 1$\left.\vphantom{ 2,2,1}\right)$ in the direction of the outward normal to the sphere x2 + y2 + z2 = 9.BITMAPSETAnswer0.2603in0.2421in0ina1

2.
Find a vector v normal to the surface z = $\sqrt{{%
x^{2}+y^{2}}}$ + $\left(\vphantom{ x^{2}+y^{2}}\right.$x2 + y2$\left.\vphantom{ x^{2}+y^{2}}\right)^{{3/2}}_{}$ at the point $\left(\vphantom{ x,y,z}\right.$x, y, z$\left.\vphantom{ x,y,z}\right)$$\left(\vphantom{ 0,0,0}\right.$0, 0, 0$\left.\vphantom{ 0,0,0}\right)$ on the surface.BITMAPSETAnswer0.2603in0.2421in0ina2

3.
Let f$\left(\vphantom{ x,y,z}\right.$x, y, z$\left.\vphantom{ x,y,z}\right)$ = ${\dfrac{{mM}}{{\sqrt{%
x^{2}+y^{2}+z^{2}}}}}$ denote Newton's gravitational potential. Show that the gradient is given by

f$\displaystyle \left(\vphantom{ x,y,z}\right.$x, y, z$\displaystyle \left.\vphantom{ x,y,z}\right)$ = - $\displaystyle {\dfrac{{mM}}{{\left( x^{2}+y^{2}+z^{2}\right) ^{3/2}}}}$$\displaystyle \left[\vphantom{
\begin{array}{c}
x \\
y \\
z
\end{array}
}\right.$$\displaystyle \begin{array}{c}
x \\
y \\
z
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{c}
x \\
y \\
z
\end{array}
}\right]$


BITMAPSETAnswer0.2603in0.2421in0ina3