(a) Prove that the expression

$$σ_{x'}σ_{y'} - τ^2_{x'y'},$$

where

$$σ_{x'}, σ_{y'}, and τ_{x'y'}$$

are components of the stress along the rectangular axes x' and y', is independent of the orientation of these axes. Also, show that the given expression represents the square of the tangent drawn from the origin of the coordinates to Mohr's circle. (b) Using the invariance property established in part a, express the shearing stress

$$τ_{xy}$$

in terms of

$$σ_x, σ_y,$$

and the principal stresses

$$σ_{max} and σ_{min}.$$

Let n be a positive integer and A the $n \times n$ matrix

$$A=\left[\begin{array}{ccccc}{1} & {\frac{1}{2}} & {\frac{1}{3}} & {\cdots} & {\frac{1}{n}} \\ {\frac{1}{2}} & {\frac{1}{3}} & {\frac{1}{4}} & {\cdots} & {\frac{1}{n+1}} \\ {\vdots} & {\vdots} & {\vdots} & {} & {\vdots} \\ {\frac{1}{n}} & {\frac{1}{n+1}} & {\frac{1}{n+2}} & {\cdots} & {\frac{1}{2 n-1}}\end{array}\right]$$

. Prove that A is positive.

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For each vector function r=r(t) (a) Find the domain. (b) Graph the curve C traced out by r=r(t), indicating its orientation. (c) Find the value of $\mathbf{r}^{\prime}(t)$ at the given number t. $\mathbf{r}(t)=3 \cos t \mathbf{i}+\sin t \mathbf{j} \quad 0 \leq t \leq \pi \quad t=\frac{\pi}{3}$

Use the finite difference method and the indicated value of n to approximate the solution of the given boundary-value problem. $y^{\prime \prime}+9 y=0, \quad y(0)=4, \quad y(2)=1 ; \quad n=4$

A sprinkler waters a circular section of lawn. a. Write an equation to represent the boundary of the sprinkler area if the endpoints of a diameter arc at (-12,16) and (12, -16). b. What is the area of the lawn that the sprinkler waters?

Who helped you address the problem? Was the salesperson empowered to give you a refund or an exchange? Or did the employee have to call in a manager?

Evaluate $\int_C \mathbf{F} \cdot d \mathbf{r}$ for the given curve. Discuss the orientation of the curve and its effect on the value of the integral.

$$\mathbf{F}(x, y)=x^2 y \mathbf{i}+x y^{3 / 2} \mathbf{j}$$

$$\mathbf{r}_2(t)=(1+2 \cos t) \mathbf{i}+\left(4 \cos ^2 t\right) \mathbf{j}, \quad 0 \leq t \leq \pi / 2$$

Use a CAS and let $\mathbf{F}(x, y, z)=x y^{2} \mathbf{i}+(y z-x) \mathbf{j}+e^{y x z} \mathbf{k}$. Use Stokes theorem to compute the surface integral of curl $\mathbf{F}$ over surface S with inward orientation consisting of cube $[0,1] \times[0,1] \times[0,1]$ with the right side missing.

Simplify the expression. Assume that all variables represent positive real numbers.

$$(-4)^{-3}$$

Simplify the expression. State the excluded values of the variables.

$$\frac{x+6}{x^2+2 x-24}$$

The torsional spring at B is undeformed when bars OB and BD are both in the vertical position and overlap. If a force F is required to position the bars at a steady orientation $\theta=60^{\circ}$, determine the torsional spring stiffness $k_T$. The slot at C is smooth, and the weight of the bars is negligible. In this configuration, the pin at C is positioned at the midpoint of the slotted bar.

Find a third-degree polynomial Q such that Q(1)=1, Q'(1)=3, Q''(1)=6, and Q'''(1)=12.

Find the eigenvalues and eigenfunctions of each of the following boundary-value problems. $y^{\prime \prime}+\lambda y=0 ; \quad y(0)-y^{\prime}(0)=0, \quad y(\pi)-y^{\prime}(\pi)=0$