If parametric equations of the flow lines are $x = x(t), y = y(t)$ what differential equations do these functions satisfy? Deduce that $dy/dx = x$.

Use implicit differentiation to find an equation of the tangent line $y^2=x-y$ at the point $(2,1)$.

For this exercise, you will differentiate implicitly to find $d y / d x$.

$x^{3 / 2}+y^{2 / 3}=1$

Use data in Appendix $\mathrm{C}$ to estimate the boiling point of benzene, $\mathrm{C}_6 \mathrm{H}_6(l)$.

How could you demonstrate boiling point?

If a wire with linear density $\rho(x, y)$ lies along a plane curve $C,$ its moments of inertia about the x- and y-axes are defined as

$$I_x = \int_Cy^2\rho(x, y)~ds \qquad I_y = \int_Cx^2\rho(x, y)~ds$$

Find the moments of inertia for the wire in the previous example.

Sketch the vector field $\text{F}(x, y) = \text{i} + x\text{j}$ and then sketch some flow lines. What shape do these flow lines appear to have?

Another example of an inverse square field is the electric force field $\text{F} = \epsilon qQ\text{r}/\lvert \text{r}\rvert^3$ discussed in the previous example. Suppose that an electron with a charge of $-1.6\times 10^{-19}$ C is located at the origin. A positive unit charge is positioned a distance $10^{-12}$ m from the electron and moves to a position half that distance from the electron. Use the previous part to find the work done by the electric force field. (Use the value  $\epsilon = 8.985\times10^9$.)

The

$$K _ { s p } \text { of } \mathrm { Ba } \left( \mathrm { IO } _ { 3 } \right) _ { 2 } \text { at } 25 ^ { \circ } \mathrm { C } \text { is } 6.0 \times 10 ^ { - 10 }$$

What is the molar solubility of

$$\mathrm { Ba } \left( \mathrm { IO } _ { 3 } \right) _ { 2 } ?$$

It is found that

$$1.1 \times 10 ^ { - 2 } \mathrm { g } \mathrm { SrF } _ { 2 }$$

dissolves per 100 mL of aqueous solution at

$$25 ^ { \circ } \mathrm { C }$$

Calculate the solubility product for

$$SrF_2.$$

An example of an inverse square field is the gravitational field $\text{F} = -(mMG)\text{r}/\lvert \text{r}\rvert^3$ discussed in the previous example. Use the previous part to find the work done by the gravitational field when the earth moves from aphelion (at a maximum distance of $1.52 \times 10^8$ km from the sun) to perihelion (at a minimum distance of $1.47 \times 10^8$ km). (Use the values $m = 5.97\times10^{24}$ kg, $M = 1.99 \times 10^{30}$ kg, and $G = 6.67\times10^{-11}$ N$\cdot$m$^2$/kg$^2$.)

Find the mass of a thin funnel in the shape of a cone $z=\sqrt{x^2+y^2}, 1 \leqslant z \leqslant 4$, if its density function is $\rho(x, y, z)=10-z$.

If the components of $\mathrm{F}$ have continuous second partial derivatives and $S$ is the boundary surface of a simple solid region, show that $\iint_S \operatorname{curl} \mathrm{F} \cdot d \mathrm{S}=0$.

Find the positively oriented simple closed curve $C$ for which the value of the line integral

$$\int_C(y^3 - y)~dx - 2x^3~dy$$

is a maximum.