Problem 10 - Calculus: Early Transcendental Functions

Question

Use the Divergence Theorem to compute \(\iint_{\partial O} \mathbf{F} \cdot \mathbf{n} d S\). $$\begin{aligned} &\begin{array}{ccccccc} Q & \text { is } & \text { bounded } & \text { by } & z=\sqrt{x^{2}+y^{2}} & \text { and } & z=4 \end{array}\\\ &\mathbf{F}=\left\langle y^{3}, x+z^{2}, z+y^{2}\right\rangle \end{aligned}$$

Step-by-Step Solution

Answer

To find the solution, we use the Divergence Theorem to express the surface integral as a volume integral. We then compute the volume integral over the region bounded by the surface, using the divergence of the vector field and the parameterization of the volume. Solve the integrals to obtain the solution.

Step 1: Identify the bounds and the function

From the problem, the bounded surface \(Q\) is determined by \(z=\sqrt{x^{2}+y^{2}}\) and \(z=4\). The vector field \(\mathbf{F}\) is given by \(\left\langle y^{3}, x+z^{2}, z+y^{2}\right\rangle\).

Step 2: Compute the Divergence of F

The divergence of \(\mathbf{F}\) is obtained by taking the divergence of \(\left\langle y^{3}, x+z^{2}, z+y^{2}\right\rangle\), which is: \(3y^{2} + 2z + 2y\).

Step 3: Parameterize volume Q

The volume Q is in cylindrical coordinates. Let \(x = r\cos(\theta), y = r\sin(\theta), z = z\). The bounds for \(r\) are \(0 \leq r \leq 4\), \(\theta\) are \(0 \leq \theta \leq 2\pi\), and \(z\) are \(r \leq z \leq 4\).

Step 4: Compute the volume integral

Use the Divergence theorem to compute the volume integral of the divergence over the volume Q. The integral is: \(\iiint_Q (3y^{2} + 2z + 2y)dV = \iiint_Q (3(r\sin(\theta))^{2} + 2z + 2(r\sin(\theta))) r dz dr d\theta\). Compute this integral.

Step 5: Solve the integrals

Solving the integrals will yield the solution to the problem. The bounds of integration were previously determined in Step 3. By solving the integral step by step, first with respect to \(z\), then \(r\), and eventually \(\theta\), the result of the surface integral is obtained.

Key Concepts

Surface Integral

A surface integral is similar to a line integral, but instead of integrating over a curve, we integrate over a surface. Think of it as adding up a certain quantity that's spread over a surface, like icing on a cake. To perform a surface integral of a vector field \textbf{F} over a surface S, we sum up the dot product of the vector field and the unit normal vector \textbf{n} to the surface, multiplied by an infinitesimal area element dS. This can be notationally expressed as \(\iint_{S} \mathbf{F} \cdot \mathbf{n} dS\). Here, each tiny part of the surface contributes to the overall integral, and geometrically, you can imagine it as the 'flow' of the vector field through the surface.

• It evaluates the cumulative effect of the vector field across a surface.
• Can represent physical concepts like fluid flow through a surface.

Vector Field

A vector field is a map that assigns a vector to every point in a subset of space. For example, imagine a weather map showing the wind velocity at different locations—that's a vector field. In mathematics, we represent a vector field by a function that takes in coordinates (such as (x, y, z) in three-dimensional space) and outputs a vector. For instance, the vector field \textbf{F} in our exercise can be written as \( \mathbf{F} = \langle y^{3}, x+z^{2}, z+y^{2} \rangle \).

• Represents phenomena where a direction and magnitude are associated with each point in space.
• Commonly used in physics to represent forces or velocities.

Divergence of a Vector Field

Divergence measures the 'outflow' of a vector field from a particular point. It's a scalar value that represents how much the vector field is spreading out (if positive) or converging in (if negative) at that point. Mathematically, for a three-dimensional vector field \textbf{F} with components (F_x, F_y, F_z), the divergence is calculated as \( abla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \). By calculating the divergence of the given vector field \(\mathbf{F}\), we establish whether it is source-like, sink-like, or neutral at each point.

• High divergence areas indicate sources, while negative divergence indicates sinks.
• It is used in various physical laws, like Gauss's law for magnetism.

Cylindrical Coordinates

Cylindrical coordinates are another way of expressing points in a three-dimensional space, just like rectangular coordinates. However, they’re particularly useful when dealing with problems that have cylindrical symmetry. A point P is given in cylindrical coordinates by \( (r, \theta, z) \) where 'r' is the radial distance from the origin, '\(\theta\)' is the angular coordinate, and 'z' is the height above the xy-plane, similar to the Cartesian 'z'. In the exercise, we convert the vector field \( \mathbf{F} \) and bounds of the volume Q into cylindrical coordinates to simplify the integration process. This approach leverages the symmetry of the problem, making complex integrals more manageable.

• Makes computations easier for symmetrical figures like cylinders and cones.
• Often simplifies integration by aligning with the natural symmetry of the problem.

Volume Integral

A volume integral extends the idea of multiple integrals into three dimensions and is used to compute integrals over a volume. When we perform a volume integral, we add up a function's values across all points in a three-dimensional space or within a specific volume. In our problem, we use the volume integral to calculate the overall divergence of the vector field \textbf{F} within the boundaries of the volume Q. The notation for a volume integral is \( \iiint_V f(x, y, z) dV \) where 'f' is the function we're integrating over and 'dV' is the infinitesimal volume element. By solving the volume integral, we determine the total quantity of whatever 'f' represents (like mass, charge, or, in our case, the divergence of a vector field), contained within the volume V.

• Can represent the total mass or charge within a three-dimensional object.
• Is crucial in fields such as fluid dynamics and electromagnetism.