For the vector field $\mathbf{D}=\hat{\mathbf{R}} 3 R^{2}$, evaluate both sides of the divergence theorem for the region enclosed between the spherical shells defined by R=1 and R=2.

Use completing the square to describe the graph of each function. Support your answers graphically. ƒ(x) = 2x² + 6x+7

Verify Stokes’s theorem for the vector field $\mathbf{A}=\hat{\mathbf{R}} \cos \theta+\hat{\phi} \sin \theta$ by evaluating it on the hemisphere of unit radius.

A single die is rolled. Find the probability of rolling an odd number or a number less than 4.

How much force would you need to cause a 20-kilogram object to accelerate in a straight line to 20 m/s^2?

Solve the equation.

$$\sqrt { x + 6 } - \sqrt { 2 x - 4 } = 0$$

If the temperature of a cake is 300°F when it leaves the oven and is 200°F ten minutes later, when will it be practically equal to the room temperature of 60°F, say, when will it be 61°F?

Determine the inverse Laplace transform $f(t)$ of the function $F(s)= \frac{5}{s^{2}+s-6}$.

If r=⟨x, y⟩, r1=⟨x1, y1⟩, and r2 = ⟨x2, y2⟩, describe the set of all points (x, y) such that |r - r1| + |r - r2| = k, where k > |r1 - r2|.

Let $f$ be a scalar field and $\mathbf{F}$ a vector field. State whether each expression is a scalar field, a vector field, or meaningless.

$$\operatorname{div}(g r a d)$$

Make an order-of-magnitude estimate of the quantity. -The width of the smallest object that can be located using sound in the range of human hearing.

Find the solution of the equation if the replacement sets are x: {1, 3, 5, 7, 9} and y: {6, 8, 10, 12, 14}. 4y = 32

Consider the scalar function $T=r \sin \theta \cos \phi$. Determine the magnitude and direction of the maximum rate of change of T at $P\left(2,6^{\circ}, 30^{\circ}\right)$.

A vector field $\mathbf{D}=\hat{\mathbf{r}} r^{3}$ exists in the region between two concentric cylindrical surfaces defined by r=1 and r=2, with both cylinders extending between z=0 and z=5. Verify the divergence theorem by evaluating the following: (a) $\oint_{S} \mathbf{D} \cdot d \mathbf{s}$ (b) $\int_{v} \nabla \cdot \mathbf{D} d v$