Vector Calculus Quiz II

 $\$  The vectors  $\overrightarrow{a},\ \ \overrightarrow{b},\ \ \overrightarrow{c}$  are coplanar iff   ________

•  $\overrightarrow{a}\times\left(\overrightarrow{b}\times\overrightarrow{c}\right)=0$  

•  $\overrightarrow{a}\times\left(\overrightarrow{b}\times\overrightarrow{c}\right)=k$  

•  $\left[\overrightarrow{a}\ \overrightarrow{b}\ \overrightarrow{c}\right]=0$  

•  $\left[\overrightarrow{a}\ \overrightarrow{b}\ \overrightarrow{c}\right]=k$  

 $\left|\overrightarrow{a}\times\overrightarrow{b}\right|$  denote ________

• the area of the parallelogram

• the volume of the parallelopiped 

• the area of the triangle

• the volume of the sphere  

 $\overrightarrow{a}.\ \left(\ \overrightarrow{b}\times\ \overrightarrow{c}\right)$  is defined by 

• Box product

• Scalar Triple Product

• The volume of the parallelopiped

• all of the above

If  $\phi$  is a scalar valued function then  $\nabla\phi$  is _______

• Scalar Valued Function

• Vector Valued function

The equation of the tangent plane to the surface  $\phi\left(x,y,z\right)=c$  

•  $\left(\overrightarrow{r}-\overrightarrow{r_o}\right)\times\nabla\phi\ =\ 0$  

•  $\left(\overrightarrow{r}-\overrightarrow{r_o}\right)\cdot\nabla\phi\ =\ 0$  

•  $\left(\overrightarrow{r}-\overrightarrow{r_o}\right)\nabla\phi\ =\ 0$  

•  $\left(\overrightarrow{r}-\overrightarrow{r_o}\right)\nabla\phi\ =\ c$  

 The equation of the normal plane to the surface  $\phi\left(x,y,z\right)=c$  

•  $\left(\overrightarrow{r}-\overrightarrow{r_o}\right)\times\nabla\phi\ =0$  

•  $\left(\overrightarrow{r}-\overrightarrow{r_o}\right)\cdot\nabla\phi\ =0$  

•  $\left(\overrightarrow{r}-\overrightarrow{r_o}\right)\nabla\phi\ =0$  

•  $\left(\overrightarrow{r}-\overrightarrow{r_o}\right)\nabla\phi\ =c$  

The equation of the tangent line to the curve which is the intersection of the two surfaces  $\phi\left(x,y,z\right)=c_1$  and  $\psi\left(x,y,z\right)=c_2$  

•  $\left(\overrightarrow{r}-\overrightarrow{r_o}\right)\times\left(\nabla\phi\times\nabla\psi\right)\ =0$  

•  $\left(\overrightarrow{r}-\overrightarrow{r_o}\right)\cdot\left(\nabla\phi\times\nabla\psi\right)\ =0$  

•  $\left(\overrightarrow{r}-\overrightarrow{r_o}\right)\cdot\left(\nabla\phi\cdot\nabla\psi\right)\ =0$  

•  $\left(\overrightarrow{r}-\overrightarrow{r_o}\right)\times\left(\nabla\phi\cdot\nabla\psi\right)\ =0$  

The curl of  $\overrightarrow{f}$  is denoted by

•  $\nabla\times\overrightarrow{f}$  

•  $\nabla\cdot\overrightarrow{f}$  

•  $\nabla\overrightarrow{f}$  

• none of the above

A vector f is called _______ if  $div\ \overrightarrow{f}=0$  

• irrotational

• Solenoidal

• Harmonic vector

• Laplacian operator

The operator  $\nabla^2$  is called

• Box product

• Square product

• Laplacian operator

• Harmonic vector