Problem 394 - Calculus Volume 3

Question

Consider \(\mathbf{F}(x, y, z)=x^{2} \mathbf{i}+x y \mathbf{j}+(z+1) \mathbf{k}\). Let \(E\) be the solid enclosed by paraboloid \(z=4-x^{2}-y^{2}\) and plane \(z=0\) with normal vectors pointing outside \(E\). Compute flux \(F\) across the boundary of \(E\) using the divergence theorem.

Step-by-Step Solution

Answer

Flux through the boundary is 40\( \pi \).

Step 1: Define the Divergence Theorem

The Divergence Theorem states that for a vector field \( \mathbf{F} \) and a closed surface \( S \) that encloses a solid volume \( E \), the flux of \( \mathbf{F} \) across \( S \) is given by \( \iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E (abla \cdot \mathbf{F}) \, dV \). This means we need to compute the divergence of \( \mathbf{F} \) and integrate it over volume \( E \).

Step 2: Compute the Divergence of \( \mathbf{F} \)

The divergence of a vector field \( \mathbf{F}(x,y,z) = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is given by \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \). For our \( \mathbf{F}(x, y, z) = x^2 \mathbf{i} + xy \mathbf{j} + (z+1)\mathbf{k} \), we have:\[ abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x^2) + \frac{\partial}{\partial y}(xy) + \frac{\partial}{\partial z}(z+1) = 2x + x + 1 = 2x + x + 1 = x + 1. \]

Step 3: Define the Region of Integration

The solid \( E \) is defined by the paraboloid \( z = 4 - x^2 - y^2 \) and the plane \( z = 0 \). It is most convenient to use cylindrical coordinates to describe this region: \( x = r\cos\theta \), \( y = r\sin\theta \), and \( z = z \), with \( 0 \leq r \leq 2 \), \( 0 \leq \theta < 2\pi \), and \( 0 \leq z \leq 4 - r^2 \).

Step 4: Convert to Cylindrical Coordinates

In cylindrical coordinates, we have \( x = r\cos\theta \), \( y = r\sin\theta \), and the volume element \( dV = r \, dr \, d\theta \, dz \). The divergence \( x + 1 = r\cos\theta + 1 \) in these coordinates. Thus, the integral becomes:\[ \iiint_E (x+1) \, dV = \int_0^{2\pi} \int_0^2 \int_0^{4-r^2} (r\cos\theta + 1)r \, dz \, dr \, d\theta. \]

Step 5: Compute the Inner Integral with Respect to \( z \)

Since there is no \( z \) in \( r\cos\theta + 1 \), the inner integral with respect to \( z \) is simply multiplying by the height \( 4 - r^2 \):\[ \int_0^{4-r^2} (r\cos\theta + 1)r \, dz = (r\cos\theta + 1)r (4 - r^2). \]

Step 6: Compute the Integral with Respect to \( r \)

Now, integrate with respect to \( r \):\[ \int_0^2 (r\cos\theta + 1)r (4 - r^2) \, dr = \int_0^2 (r^2\cos\theta + r)(4 - r^2) \, dr. \]Expanding: \[ = \int_0^2 (4r^2\cos\theta - r^4\cos\theta + 4r - r^3) \, dr. \]Compute each term separately and combine results.

Step 7: Compute the Integral with Respect to \( \theta \)

After computing the second integral over \( r \), integrate the result with respect to \( \theta \) from \( 0 \) to \( 2\pi \). Note that the terms involving \( \cos\theta \) will integrate to zero over this symmetric interval. Apply the integral results from Step 6.

Step 8: Final Evaluation and Result

Evaluate the values from Step 7 for each term. Any terms that disappear due to symmetry or simplification should be noted, leaving the final numerical result. This result gives the total flux across the boundary of \( E \).

Key Concepts

Flux Calculation

Flux calculation is a vital concept used to determine how much of a vector field passes through a given surface. It essentially involves calculating the surface integral of the field. In this exercise, we deal with the flux of the vector field \( \mathbf{F}(x, y, z) = x^2 \mathbf{i} + xy \mathbf{j} + (z+1) \mathbf{k} \). To find the flux across the boundary of region \( E \), the Divergence Theorem is applied.The Divergence Theorem links the flux of a vector field across a closed surface to the divergence over the volume enclosed by the surface. In formula terms, it says:

• \( \iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E (abla \cdot \mathbf{F}) \, dV \)

This translates to computing the triple integral of the divergence of \( \mathbf{F} \) over the volume \( E \).The benefit of this theorem is that it often simplifies the calculation, especially in complex regions. Instead of directly computing a possibly cumbersome surface integral, you shift the focus to the volume, which can be more manageable, especially when using coordinates that fit the symmetry of the region.

Cylindrical Coordinates

Cylindrical coordinates are a spatial coordinate system that is especially useful for dealing with problems involving circular symmetry, like this one involving a paraboloid. They are a generalization of polar coordinates into three dimensions (adding the \( z \)-axis), and they help express points in space using three parameters: \( r \), \( \theta \), and \( z \).In this problem, the region \( E \) enclosed by the paraboloid \( z = 4 - x^2 - y^2 \) and the plane \( z = 0 \), is naturally suited for cylindrical coordinates:

• \( x = r\cos\theta \)
• \( y = r\sin\theta \)
• \( z = z \)

This helps describe \( E \) with ranges: \( 0 \leq r \leq 2 \), \( 0 \leq \theta < 2\pi \), and \( 0 \leq z \leq 4 - r^2 \).Cylindrical coordinates simplify the integration for volumes with rotational symmetry. The conversion allows the volume element \( dV \) to be expressed as \( r \, dr \, d\theta \, dz \), incorporating the radial distance, which compensates for the circular base shape.

Vector Field Divergence

In vector calculus, the divergence of a vector field is a scalar quantity that represents how much the vector field expands or contracts at a point. You can think of it as measuring the field's tendency to originate from or converge into a specific point.The divergence is calculated as the dot product of the del operator \( abla \) with the vector field \( \mathbf{F} \). For a field \( \mathbf{F}(x, y, z) = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), its divergence is given by:

• \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \)

Applying this to \( \mathbf{F}(x, y, z) = x^2 \mathbf{i} + xy \mathbf{j} + (z+1)\mathbf{k} \), we got the divergence:

• \( abla \cdot \mathbf{F} = 2x + x + 1 = x + 1 \)

In the context of the Divergence Theorem, calculating the divergence lets us substitute this value into the integral over the volume \( E \). This calculation is crucial because it ultimately simplifies to finding the total flux across the boundary of \( E \).