Problem 410 - Calculus Volume 3

Question

Let \(E\) be the solid bounded by the \(x y\) -plane and paraboloid \(z=4-x^{2}-y^{2}\) so that \(S\) is the surface of the paraboloid piece together with the disk in the \(x y\) -plane that forms its bottom. \(\mathbf{F}(x, y, z)=\left(x z \sin (y z)+x^{3}\right) \mathbf{i}+\cos (y z) \mathbf{j}+\left(3 z y^{2}-e^{x^{2}+y^{2}}\right) \mathbf{k}\) find \(\int / \mathbf{F} \cdot d \mathbf{S}\) using the divergence theorem.

Step-by-Step Solution

Answer

The surface integral \(\int \mathbf{F} \cdot d\mathbf{S}\) is \(192\pi\).

Step 1: Understand the problem

We need to find the surface integral \(\int \mathbf{F} \cdot d\mathbf{S}\) where \(\mathbf{F}\) is a vector field, and \(S\) is the surface of the solid \(E\) defined by the paraboloid \(z=4-x^2-y^2\) and the disk in the \(xy\)-plane. We will use the divergence theorem to simplify the calculation.

Step 2: State the Divergence Theorem

The Divergence Theorem states: \[ \int_S \mathbf{F} \cdot d\mathbf{S} = \int_V (abla \cdot \mathbf{F}) \, dV \]where \(V\) is the volume enclosed by \(S\), and \(abla \cdot \mathbf{F}\) is the divergence of \(\mathbf{F}\). We need to first calculate the divergence of \(\mathbf{F}\).

Step 3: Calculate the divergence \(\nabla \cdot \mathbf{F}\)

Given \(\mathbf{F}(x, y, z) = (xz \sin(yz) + x^3) \mathbf{i} + \cos(yz) \mathbf{j} + (3zy^2 - e^{x^2 + y^2}) \mathbf{k}\), we find its divergence as follows:1. The partial derivative with respect to \(x\) of the \(i\)-component:\(\frac{\partial}{\partial x} (xz \sin(yz) + x^3) = z \sin(yz) + 3x^2\)2. The partial derivative with respect to \(y\) of the \(j\)-component:\(\frac{\partial}{\partial y} \cos(yz) = -z \sin(yz)\)3. The partial derivative with respect to \(z\) of the \(k\)-component:\(\frac{\partial}{\partial z} (3zy^2 - e^{x^2 + y^2}) = 3y^2\)Combining these, we find:\[ abla \cdot \mathbf{F} = z \sin(yz) + 3x^2 - z \sin(yz) + 3y^2 = 3x^2 + 3y^2 \]

Step 4: Set up the volume integral

Now we need to evaluate the integral \(\int_V (3x^2 + 3y^2) \, dV\) over the volume \(V\) enclosed by the surface of the solid \(E\). Since \(E\) is bounded by \(z=4-x^2-y^2\), the limits for \(z\) are from \(0\) to \(4-x^2-y^2\). In the \(xy\)-plane, the solid is a disk with radius 2 (since when \(z=0\), \(4 = x^2 + y^2\)), in polar coordinates.

Step 5: Convert to polar coordinates

In polar coordinates, define \(x = r \cos \theta\) and \(y = r \sin \theta\). Then the integral becomes:\[ \int_0^{2\pi} \int_0^2 \int_0^{4-r^2} (3r^2) \, dz \, r \, dr \, d\theta \] The term \(3r^2\) comes from \(3(x^2 + y^2) = 3r^2\), and \(r\) is the additional factor from converting \(dV\) into polar terms \(r \, dr \, d\theta\).

Step 6: Evaluate the integral

Firstly, integrate with respect to \(z\):\[ \int_0^{4-r^2} 3r^2 \, dz = 3r^2 \cdot z \Big|_0^{4-r^2} = 3r^2(4-r^2) = 12r^2 - 3r^4 \]Next, integrate with respect to \(r\):\[ \int_0^2 (12r^2 - 3r^4) \, r \, dr = \int_0^2 (12r^3 - 3r^5) \, dr \]This becomes:\[ \int_0^2 12r^3 \, dr = 3r^4 \Big|_0^2 = 192 \, ext{and} \, \int_0^2 3r^5 \, dr = r^6 \Big|_0^2 = 96 \]Combine these to get:\[ 192 - 96 = 96 \]Finally, integrate with respect to \(\theta\):\[ \int_0^{2\pi} 96 \, d\theta = 96\theta \Big|_0^{2\pi} = 192\pi \]

Step 7: Conclude

Using the divergence theorem, the surface integral \(\int \mathbf{F} \cdot d\mathbf{S}\) is equal to the volume integral calculated, which evaluates to \(192\pi\).

Key Concepts

Vector Calculus

Vector calculus is an essential field of mathematics that deals with vector fields and differential equations. It plays a vital role in physics and engineering by providing tools to describe physical phenomena mathematically. Key operations in vector calculus include:

• Gradient — measures how a scalar field changes across space.
• Divergence — measures a vector field's tendency to originate from or converge into a point.
• Curl — measures the rotation or swirling of a vector field.

The Divergence Theorem, part of vector calculus, connects the flow of a vector field through a closed surface to the behavior of the vector field inside the volume enclosed by that surface. Understanding these concepts allows mathematicians and engineers to solve complex spatial and dynamic problems efficiently.

Surface Integral

Surface integrals are used to evaluate the total of some quantity over a curved surface, taking into account the vector field impacting the surface. They apply especially in physics for calculating quantities like flux across a surface. In this context, given a vector field \( \mathbf{F} \), the surface integral \( \int_S \mathbf{F} \cdot d\mathbf{S} \) signifies how much of \( \mathbf{F} \) penetrates or passes through the surface \( S \).
To compute a surface integral, it typically involves:

• Parametrizing the surface to relate it to a known coordinate system.
• Computing the dot product of the vector field with the surface's normal vector.
• Integrating this product over the entire surface.

In the divergence theorem scenario, the complexity of directly calculating a surface integral can be circumvented by converting it to a volume integral over the solid bounded by the surface, which can be simpler to evaluate.

Paraboloid

A paraboloid is a three-dimensional surface that can be defined by revolving a parabolic curve around its axis of symmetry. In mathematical terms, it can be expressed in a standard form, such as \( z = a - x^2 - y^2 \). For this exercise, the paraboloid given by \( z = 4 - x^2 - y^2 \) forms a solid shape when bounded by the \( xy \)-plane.
The importance of understanding the paraboloid in this scenario includes:

• Visualizing the boundary for integration when applying the divergence theorem.
• Identifying the limits of integration for computing the volume integral.
• Ensuring the correct setup of the coordinate transformations, such as switching to polar coordinates where needed.

Recognizing the geometry of paraboloids aids in solving integral problems and anticipating how they impact vector fields.

Vector Field

A vector field associates a vector to every point in space, thereby describing how vectors vary in a given area. In physics, vector fields can represent various force fields, such as electric or magnetic fields, or fluid flow. The given vector field \( \mathbf{F}(x, y, z) = \left(xz \sin(yz) + x^3\right) \mathbf{i} + \cos(yz) \mathbf{j} + \left(3zy^2 - e^{x^2 + y^2}\right) \mathbf{k} \) has distinct components for each spatial direction.
When working with a vector field:

• We compute partial derivatives to find the divergence (\( abla \cdot \mathbf{F} \)), indicating how much the field is spreading out or gathering.
• Use divergence to transform surface integrals into volume integrals, via the divergence theorem.
• Visualizing the vector field's overall behavior is helpful in confirming mathematical results with physical intuition.

Understanding vector fields in-depth enables the application of various calculus techniques to solve practical problems in science and engineering.