User:Bartlete2/EGM6611Homework1

=Homework 1=

Problem Statement
Part (a). Find $${\vec{u}}{\cdot}{\vec{v}}$$, $${\vec{u}}{\times}{\vec{v}}$$, and $${\vec{v}}{\times}{\vec{u}}$$.

Part (b). Find the projection of $${\vec{c}}$$ onto $${\vec{d}}$$ where $$({\vec{i}},{\vec{j}},{\vec{k}})$$ is an orthonormal set.

Part a
By definition the dot product expressed in Einstein notation is:

And the cross product is:

Using the anticommutative property of the cross product:

Part b
By definition the projection of c onto d is:

The magnitude of d is:

And again using the definition of the scalar (dot) product:

And the product of a vector and the scalar product is

Problem Statement
Write out the summation implied by the repeated indices where m,n,k=1..3.

Solution
$${\delta}_{mn}$$ is the kronecker delta function defined as:

{{NumBlk|::|$${\delta}_{mn}=\{_{0 if m{\not=}n}^{1 if m=n}$$ |$$|}}

Therefore equation 2-1 can be written out as follows:

Similarly, equations 2-2 through 2-4 can be written out as:

Problem Statement
Write out the summation implied by the repeated indices where m,n,k,r,s=1..3.

Problem Statement
The orthonormal basis $$(e_{1}',e_{2}',e_{3}')$$ is obtained by a rotation through an angle$${\theta}$$ about the $$x_{3}$$ axis. Give the components in the new basis of the vector.

Solution
Since the transformation matrix is orthogonal:

Problem Statement
Given the tensor, A, show that there exists a non-zero vector n such that An=0 if and only if the determinant of A is zero. $$ A= \begin{pmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \\ \end{pmatrix} $$

Solution
Using Cramer's Rule:

$$ \begin{vmatrix} 0 & A_{12} & A_{13} \\ 0 & A_{22} & A_{23} \\ 0 & A_{32} & A_{33} \\ \end{vmatrix}= \begin{pmatrix} n_{1} \\ n_{2} \\ n_{3} \\ \end{pmatrix} $$
 * A|

By the properties of determinants, if a column is zero, the determinant is zero. Therefore if $${\vec{n}}{\not=}0$$ then $$|A|=0$$.

Given Information
$$ T = \begin{pmatrix} 5 & -1 & -1 \\ -1 & 4 & 0 \\  -1 & 0 & 4  \\ \end{pmatrix} $$

Problem Statement
(a). Show that the principle directions of the second-order tensor T coincide with that of T2. (b). If T has the matrix representation with respect to an orthonormal basis, obtain the matrix representation of $$\sqrt{T}$$ with respect to the same basis.

Solution
An eigenvalue is computed from the following equation:

If the new vectors are again multiplied by $$\mathbf{A}$$ and $${\lambda}$$,

The eigenvalue is squared, but the eigenvector, or principle direction, is unchanged.

To find the square root, the following diagonalizing relationship was used:

Where D is the diagonal matrix of eigenvalues and X is a matrix whose columns are eigenvectors of T.

And the inverse of X is:

Problem Statement
Determine the derivative of the function in the direction of the unit normal $$n=\frac{1}{\sqrt{3}}(1,1,1)$$ at the point (2,-1,0).

Solution
The gradient of F is:

The directional derivative is: