10 Parametric Surfaces and Surface Integrals Free Online FlashCards

Why do parametrized surfaces generally use two parameters?

A surface is two-dimensional, so its points generally require two parameters, such as $u$ and $v$, in a vector-valued parametrization $\mathbf r(u,v)$.

What is the parameter domain?

The parameter domain is the region $D$ in the $uv$-plane over which the parameters vary to generate the surface.

How are coordinate curves obtained from $\mathbf r(u,v)$?

Holding $u$ constant and varying $v$ gives one family of coordinate curves; holding $v$ constant and varying $u$ gives the other.

What condition makes a parametrization regular?

The parametrization is regular at a point when $\mathbf r_u\times\mathbf r_v\ne\mathbf 0$. Then the tangent directions are not parallel and determine a tangent plane.

How is the tangent plane found from a parametrization?

At $P=\mathbf r(u_0,v_0)$, a tangent-plane equation is $(\mathbf r_u\times\mathbf r_v)|_{(u_0,v_0)}\cdot\langle x-x_0,y-y_0,z-z_0\rangle=0$.

What are the two unit normals to an oriented parametrized surface?

The two unit normals are $\mathbf N=\pm\frac{\mathbf r_u\times\mathbf r_v}{\|\mathbf r_u\times\mathbf r_v\|}$. Choosing the sign fixes the surface orientation.

How is the graph $z=f(x,y)$ parametrized?

A graph $z=f(x,y)$ can be parametrized by $\mathbf r(x,y)=\langle x,y,f(x,y)\rangle$ over its region $D$.

What is the surface-area element for a parametrized surface?

For $\mathbf r(u,v)$, the surface-area element is $dS=\|\mathbf r_u\times\mathbf r_v\|\,du\,dv$.

What is the area formula for a graph?

For $z=f(x,y)$, $dS=\sqrt{1+f_x^2+f_y^2}\,dA$, so $\operatorname{Area}(S)=\iint_D\sqrt{1+f_x^2+f_y^2}\,dA$.

How is a scalar surface integral computed parametrically?

A scalar surface integral is $\iint_S g\,dS=\iint_D g(\mathbf r(u,v))\|\mathbf r_u\times\mathbf r_v\|\,du\,dv$.

What physical quantities can a scalar surface integral represent?

If $g$ is surface density, then $\iint_S g\,dS$ gives total mass. If $g=1$, it gives the surface area.

What is the parametrized formula for flux?

The flux is $\iint_S\mathbf F\cdot\mathbf N\,dS=\iint_D\mathbf F(\mathbf r(u,v))\cdot(\mathbf r_u\times\mathbf r_v)\,du\,dv$.