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Next: Chapters Up: VECTORS, TENSORS AND MATRIX Previous: Questions Chapters
- A.3
-
For the following vectors calculate
, , and . Show the calculations by writing and in matrix form.
- A.4
-
Consider the vectors
and
- a)
- Resolve
into two vectors: one in the direction of and the other normal to .
- b)
- Sketch
, and the two components of found above.
- c)
- Determine the dot product of the two components of
.
- d)
- Find a vector
that is perpendicular to both and .
- A.5
-
For the following questions use the vectors
, , and . Find:
- a.)
![$\mathbf{a}\cdot\mathbf{b}$](img24.png)
- b.)
![$\mathbf{b}\times\mathbf{c}$](img90.png)
- c.)
![$\mathbf{c}\otimes\mathbf{b}$](img91.png)
- d.)
![$\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})$](img92.png)
- e.)
![$(\mathbf{a}\times\mathbf{c})\times(\mathbf{b}\times\mathbf{c})$](img93.png)
- f.)
![$\mathbf{a}\times(\mathbf{b}+\mathbf{c})$](img94.png)
- g.)
![$(\mathbf{a}\times\mathbf{c})+(\mathbf{b}\times\mathbf{c})$](img95.png)
- A.6
-
For the velocity vectors
, and , determine the following:
- a)
- The divergence of
: ![$\nabla\cdot\mathbf{v}$](img78.png)
- b)
- The curl of
: ![$\nabla\times\mathbf{v}$](img79.png)
- c)
![$\mathbf{v}\cdot(\nabla\mathbf{v})$](img81.png)
- d)
![$(\mathbf{v}\cdot\nabla)\mathbf{v}$](img82.png)
- A.7
-
For the vector-valued functions of position
and (velocities) and for the scalar-valued function of position (density), prove the following identities. Use a rectangular coordinate representation for , and (e.g., ) to compute the expressions on each side of each equation. Verify using Scientific Workplace.
- a)
![$\nabla\cdot\left(\rho\mathbf{v}\right)=\mathbf{v}\cdot\nabla\rho+\rho\nabla\cdot\mathbf{v}$](img103.png)
- b)
![$\nabla\times\left(\rho\mathbf{v}\right)=\nabla\rho\times\mathbf{v}+\rho\nabla\times\mathbf{v}$](img104.png)
- c)
![$\nabla\cdot\mathbf{u}\times\mathbf{v})=(\nabla\times\mathbf{u})\cdot\mathbf{v} -\mathbf{u}\cdot\nabla\times\mathbf{v})$](img105.png)
- d)
![$\nabla\cdot(\nabla\times\mathbf{u})=0$](img106.png)
- A.8
- Show that
, where is a scalar-valued function of position, i.e., .
- A.9
-
Given the matrices:
evaluate the following:
- (a)
![$[B]+[D]$](img111.png)
- (b)
![$[A][A]^T$](img112.png)
- (c)
![$[A][B]$](img113.png)
- (d)
![$[C][A][B]$](img114.png)
- (e)
![$[C][D]$](img115.png)
- (f)
![$[D][C]$](img116.png)
- A.10
-
Given the following set of linear equations in matrix form, write the equations as three separate equations.
- A.11
-
Write the following linear algebraic equations in matrix form:
- A.12
-
Solve for
from the set of equations
Next: Chapters Up: VECTORS, TENSORS AND MATRIX Previous: Questions Chapters
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