Problem 26 - Thomas Calculus

Question

Maximum flux Among all rectangular solids defined by the inequalities \(0 \leq x \leq a, 0 \leq y \leq b, 0 \leq z \leq 1,\) find the one for which the total flux of \(F=\left(-x^{2}-4 x y\right) \mathbf{i}-6 y z \mathbf{j}+12 z \mathbf{k}\) outward through the six sides is greatest. What is the greatest flux?

Step-by-Step Solution

Answer

The greatest flux is 4.

Step 1: Understanding the Flux through a Surface

The flux of a vector field \( F \) through a closed surface \( S \) is given by the surface integral \( \iint_S F \cdot d\mathbf{S} \). For a closed surface in three dimensions, this can be evaluated using the Divergence Theorem: \( \iiint_V (abla \cdot F) \, dV \), where \( V \) is the volume enclosed by \( S \).

Step 2: Calculate the Divergence of the Vector Field

First, we find the divergence of the vector field \( F(x, y, z) = (-x^2 - 4xy)i - 6yzj + 12zk \). The divergence is \( abla \cdot F = \frac{\partial}{\partial x}(-x^2 - 4xy) + \frac{\partial}{\partial y}(-6yz) + \frac{\partial}{\partial z}(12z) = -2x - 4y - 6y + 12 = -2x - 10y + 12 \).

Step 3: Set Up the Triple Integral

To find the total flux, we compute the integral of the divergence over the volume defined by \( 0 \leq x \leq a \), \( 0 \leq y \leq b \), and \( 0 \leq z \leq 1 \). The integral is: \[\iiint_V (-2x - 10y + 12) \, dV = \int_0^1 \int_0^b \int_0^a (-2x - 10y + 12) \, dx \, dy \, dz.\]

Step 4: Evaluate the Integral with Respect to \( x \)

Integrating with respect to \( x \) first, we have: \[\int_0^a (-2x - 10y + 12) \, dx = \left[ -x^2 - 10yx + 12x \right]_0^a = -a^2 - 10ay + 12a.\]

Step 5: Evaluate the Integral with Respect to \( y \)

Now integrate with respect to \( y \): \[\int_0^b (-a^2 - 10ay + 12a) \, dy = \left[-a^2y - 5ay^2 + 12ay \right]_0^b = -a^2b - 5ab^2 + 12ab.\]

Step 6: Evaluate the Integral with Respect to \( z \)

Finally, integrate with respect to \( z \): \[\int_0^1 (-a^2b - 5ab^2 + 12ab) \, dz = -a^2b - 5ab^2 + 12ab.\]

Step 7: Maximize the Expression for Flux

To find the values of \( a \) and \( b \) that maximize the flux, consider the function \( f(a, b) = -a^2b - 5ab^2 + 12ab \). Taking partial derivatives and setting them to zero gives critical points: \(\frac{\partial f}{\partial a} = -2ab - 5b^2 + 12b = 0 \) and \( \frac{\partial f}{\partial b} = -a^2 - 10ab + 12a = 0 \).Solving these equations, we find that the rectangular solid which maximizes the flux is defined by \( a = 2 \) and \( b = \frac{2}{5} \).

Step 8: Calculate the Greatest Flux

Substitute \( a = 2 \) and \( b = \frac{2}{5} \) into the flux expression: \[-2^2 \times \frac{2}{5} - 5 \times 2 \times \left( \frac{2}{5} \right)^2 + 12 \times 2 \times \frac{2}{5} = -\frac{8}{5} - \frac{20}{25} + \frac{48}{5} = 4.\] The greatest flux is therefore 4.

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields and their differentiation and integration. It extends calculus to vector functions and provides powerful tools for physics and engineering.
In vector calculus, you often deal with concepts like gradients, divergences, and curls. These are all ways of measuring different properties of vector fields.

• The **gradient** gives you a vector that represents the direction and rate of fastest increase of a function.
• The **divergence** tells you how much a vector field spreads out from a point.
• The **curl** measures the tendency of a vector field to rotate around a point.

Understanding these operations is essential when investigating the behavior of a vector field and how it interacts with spaces and surfaces. They bridge the gap between calculus and physical phenomena.

Divergence Theorem

The Divergence Theorem is a crucial result in vector calculus. It links the divergence of a vector field within a volume to the flux across the surface bounding that volume.
Simply put, it helps you evaluate the total 'outflow' from a volume by considering how much the field 'diverges' at each point within it.
Mathematically, it states\[\iint_{S} \mathbf{F} \cdot d\mathbf{S} = \iiint_{V} (abla \cdot \mathbf{F}) \, dV\]where \( \mathbf{F} \) is a vector field, \( S \) is a closed surface, and \( V \) is the volume bounded by \( S \). This offers a neat way to compute surface integrals by converting them to volume integrals.
The theorem is particularly useful in simplifying complex surface calculations into more manageable volume problems, widely applied in electromagnetism, fluid dynamics, and engineering.

Vector Field

A vector field assigns a vector to every point in a space. It's like having arrows attached to each point, showing direction and magnitude.
In physics, vector fields represent phenomena like force fields, velocity fields, or electromagnetic fields.
In our exercise, the vector field is given by:\[\mathbf{F}(x, y, z) = (-x^2 - 4xy) \mathbf{i} - 6yz \mathbf{j} + 12z \mathbf{k}\]Each component of \( \mathbf{F} \) depends on coordinates \( x, y, \) and \( z\), representing different effects in each direction.

• The \( x\)-component affects how much the field points in the \( \mathbf{i}\) direction.
• The \( y\)-component influences the \( \mathbf{j}\) direction.
• The \( z\)-component determines the flow towards the \( \mathbf{k}\) axis.

Understanding vector fields is vital for predicting how they behave over space, helping you model and analyze real-world dynamics.

Surface Integral

Surface integrals in vector calculus help measure the extent of a vector field passing through a surface. They are broader than conventional integrals, allowing you to calculate quantities over curved surfaces.
If you think of a surface in a vector field, its interaction is described by adding up all the field's 'flow' across it. This concept is fundamental in physics, where these integrals compute flux in electromagnetism or fluid flow.
The mathematical form is given by:\[\iint_{S} \mathbf{F} \cdot d\mathbf{S}\]Here, \( d\mathbf{S} \) represents an infinitesimal area of the surface \( S \), with a vector normal to the surface.

• **Flow direction:** Dependent on \(d\mathbf{S}\), pointing outward.
• **Strength of flow:** Captured by \(\mathbf{F} \cdot d\mathbf{S}\).

Surface integrals can be challenging, but with the Divergence Theorem, they often simplify to volume integrals, providing an elegant computational shortcut.