Section 6 Wrap Up Questions
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Are the following statements true or false? Give reasons for your answers.
- The integral \(\int \int_{S} \left( x^2\vec{i} +y^2\vec{j} +z^2\vec{k} \right)\cdot d\vec{S}\) is always positive since the components of \(x^2\vec{i} +y^2\vec{j} +z^2\vec{k}\) are always positive.
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If \(S\) is the full sphere \(x^2+y^2+z^2=1\) and \(D\) is the upper half portion of that sphere, then
\begin{equation*} \int \int_{S} \vec{F} (x,y,z)\cdot d\vec{S}=2\int \int_{D} \vec{F} (x,y,z)\cdot d\vec{S}\text{.} \end{equation*} - The integral \(\int \int_{S} \vec{F} (x,y,z)\cdot d\vec{S}\) is largest when the vectors in \(\vec{F}\) point perpendicular to the orientation on \(S\text{.}\)
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If \(-S\) denotes the surface \(S\) with opposite orientation, then for any vector field \(\vec{F}\text{,}\) we have
\begin{equation*} \int \int_{-S} \vec{F} (x,y,z)\cdot d\vec{S} =-\int \int_{S} \vec{F} (x,y,z)\cdot d\vec{S}\text{.} \end{equation*}
- Explain why even though a surface \(S\) lives in \(3\)-space, the integral \(\int \int_{S} \vec{F} (x,y,z)\cdot d\vec{S}\) is a double integral and not a triple integral.
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Recall that a line integral of a vector field \(\vec{F}\) over a curve \(C\) parameterized by \(\vec{r}(t)\) for \(a\leq t\leq b\) is calculated using the formula
\begin{equation*} \int_{C} \vec{F} \cdot d\vec{S}=\int_{a}^{b} \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t)\,dt. \end{equation*}Describe the differences and similarities to the surface integral \(\int \int_{S} \vec{F} (x,y,z)\cdot d\vec{S}\text{.}\) That is, describe how the formulas differ or share similarities and how the process of calculation differs or share similarities.