Show that Lagrange's method automatically yields the correct equations of motion for a particle moving in a plane in a rotating coordinate system Oxy. (Hint: $T=\frac{1}{2} m \mathbf{v} \cdot \mathbf{v}$, where $\mathbf{v}=\mathbf{i}(\dot{x}-\omega y)+\mathbf{j}(\dot{y}+\omega x), \text { and } F_{x}=-\partial \mathrm{V} / \partial x, F_{y}=-\partial \mathrm{V} / \partial y$.)

What is an $n$th partial sum?

Let

$$\mathbf{F}=\frac{y}{x^2+y^2} \mathbf{i}-\frac{x}{x^2+y^2} \mathbf{j}=M \mathbf{i}+N \mathbf{j}$$

(a) Show that $\partial N / \partial x=\partial M / \partial y$.

(b) Show, by using the parametrization $x=\cos t, y=\sin t$, that $\oint_C M d x+N d y=-2 \pi$, where C is the unit circle.

(c) Why doesn't this contradict Green's Theorem?

Suppose a movie director is filming on a set that should look like a hot desert. He wants the scene to appear warmer, such that the red and yellow tones are the most apparent. What color filters should he use? What color will the blue sky appear when he uses those filters, and why?

In view of Clapeyron's equation and figure, is there something special about ice $I$ versus the other forms of ice ?

Draw and Label the bones and features indicated on the anterior view of the Right knee , using the terms provided
Tibia

Write the rational function given as a partial sum of partial fractions.

$$\frac{4 x^{3}+4 x^{2}+x-1}{x^{2}(x+1)^{2}}$$

Suppose that $f$ is differentiable at $\left(x_0, y_0\right)$ and $\nabla f\left(x_0, y_0\right) \neq$ $\mathbf{0}$. Calculate the rate of change of $f$ in the direction of

$$\frac{\partial f}{\partial y}\left(x_0, y_0\right) \mathbf{i}-\frac{\partial f}{\partial x}\left(x_0, y_0\right) \mathbf{j} \text {. }$$

Give a geometric interpretation to your answer.

In an electric circuit with two resistors of resistance $R_1$ and $R_2$ connected in parallel, the total resistance R is given by the formula $\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}.$ Show that $R_{1} \frac{\partial R}{\partial R_{1}}+R_{2} \frac{\partial R}{\partial R_{2}}=R$

Hot air at $80^{\circ} \mathrm{C}$ is blown over a $2-m \times 4-m$ flat surface at $30^{\circ} \mathrm{C}.$ If the average convection heat transfer coefficient is $55 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K},$ determine the rate of heat transfer from the air to the plate, in kW.

Calculate the partial derivatives.

$$f(x, y, z)=z^{x y^2}$$

A conducting sphere of radius c is grounded and placed in a uniform electric field that has intensity E in the z-direction. The potential $u(r, \theta)$ outside the sphere is determined from the boundary-value problem $\frac{\partial^{2} u}{\partial r^{2}}+\frac{2}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}+\frac{\cot \theta}{r^{2}} \frac{\partial u}{\partial \theta}=0, \quad r>c, 0<\theta<\pi$

$$\begin{array}{l}{u(c, \theta)=0, \quad 0<\theta<\pi} \\ {\lim _{r \rightarrow \infty} u(r, \theta)=-E z=-E r \cos \theta}\end{array}$$

. Show that $u(r, \theta)=-E r \cos \theta+E \frac{c^{3}}{r^{2}} \cos \theta$.

Ms. Augello is making tie dye shirts with her students. Each gallon of hot water needs

$$\frac{2}{3}$$

cup of tie dye. If Ms. Augello has

$$5\frac{1}{4}$$

cups of tie dye, how many batches of solution will she be able to make?

Find the partial derivatives. Assume the variables are restricted to a domain on which the function is defined.

$$\frac{\partial}{\partial w}\left(\frac{x^2 y w-x y^3 w^7}{w-1}\right)^{-7 / 2}$$