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\noindent {\bf 21-259} \hfill {\bf Calculus in 3-D} \hfill {\bf Spring 2003}

\centerline{\bf Formula Sheet for Test 3} 

\vspace{.60in}

\noindent {\bf Cylindrical Coordinates}: 

$$x = r \ cos \theta,\ \ y = r \sin \theta,\ \ z = z, \ \ r = \sqrt{x^2+y^2}, \ \ dV = r \ dzdrd\theta$$ 

\vspace{.60in} 

\noindent {\bf Spherical Coordinates}:
$$x = \rho \sin \varphi \cos \theta, \ \ y = \rho \sin \varphi \sin \theta,\ z = \rho \cos \ \varphi , \ \ \ \ \rho = \sqrt{x^2+y^2+z^2}, \ \ dV = \rho^2 \sin \varphi\ d \rho d \varphi d \theta$$


\vspace{.60in} 

\noindent {\bf Green's Theorem:} 

$$\int_C P(x,y) dx + Q(x,y) dy = \int_D \int \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA $$ 

\noindent Here $C$ is a simple closed curve in the $x-y$ plane with counterclockwise orientation and $D$ is the region enclosed by $C$. 

\vspace{.60in} 

\noindent {\bf Surfaces Described by $\vec{r} (u,v)$:} 

$$dS = \left|\frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}\right| dudv, \ \ \ \ \vec{n}dS = \pm \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} dudv$$

\vspace{.60in} 

\noindent {\bf Surfaces Described by $z=f(x,y)$:} 

$$dS = \sqrt{1+f^2_x+f^2_y}\  dydx, \ \ \ \ \vec{n}dS = \pm \left(f_x \vec{i}+f_y\vec{j}-\vec{k}\right) dydx$$

\vspace{.60in} 

\noindent {\bf Sphere of Radius $a>0$ Centered at $(0,0,0)$:} 

$$\vec{r} (\varphi,\theta) = (a \sin \varphi \cos \theta) \vec{i} + (a \sin \varphi \sin \theta) \vec{j} + (a \cos \varphi) \vec{k}\  \ dS = a^2 \sin \varphi d \varphi d \theta$$

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