Problem 44 - Thomas Calculus

Question

Maximizing work Among all smooth, simple closed curves in the plane, oriented counterclockwise, find the one along which the work done by $$\mathbf{F}=\left(\frac{1}{4} x^{2} y+\frac{1}{3} y^{3}\right) \mathbf{i}+x \mathbf{j}$$ is greatest. \((\)Hint: Where is \((\operatorname{curl} \mathbf{F}) \cdot \mathbf{k}\) positive?\()\)

Step-by-Step Solution

Answer

The work is maximized along the ellipse \( \frac{1}{4} x^2 + y^2 = 1 \), oriented counter-clockwise.

Step 1: Identify the Mathematical Tools Needed

To solve this problem, we will utilize Green's Theorem, which relates a line integral around a simple closed curve to a double integral over the plane region bounded by the curve. Green's Theorem is particularly useful because it turns a line integral problem into a potentially simpler area integral problem.

Step 2: Apply Green's Theorem

Green's Theorem states that for a vector field \( \mathbf{F} = M \mathbf{i} + N \mathbf{j} \), the line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) is equal to the double integral \( \iint_R (\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}) \ dA \), where \( R \) is the region inside curve \( C \).

Step 3: Compute the Curl of \( \mathbf{F} \)

Given \( \mathbf{F} = \left(\frac{1}{4} x^{2} y + \frac{1}{3} y^{3}\right) \mathbf{i} + x \mathbf{j} \), compute \( abla \times \mathbf{F} \). This is \( \operatorname{curl} \mathbf{F} = \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \). With \( N = x \) and \( M = \frac{1}{4} x^{2} y + \frac{1}{3} y^{3} \), we find \( \frac{\partial N}{\partial x} = 1 \) and \( \frac{\partial M}{\partial y} = \frac{1}{4} x^{2} + y^2 \).

Step 4: Set Up the Double Integral

The double integral becomes \( \iint_R \left(1 - (\frac{1}{4} x^{2} + y^2)\right) \ dA \). To maximize the work done, we need \( 1 - (\frac{1}{4} x^2 + y^2) > 0 \). This is a constraint on the region \( R \).

Step 5: Identify the Region

To maximize the integral, the point is to maximize the area \( R \) where the integrand is positive. The inequality \( \frac{1}{4} x^2 + y^2 < 1 \) is the region inside an ellipse or an ellipse-like shape, determined by solving the equation. Thus, the curve maximizing the integral encloses this region.

Step 6: Conclude the Solution

The region \( R \) must be chosen such that it fits within the constraints derived from the inequality. The maximum work is done along the curve enclosing the ellipse defined by \( \frac{1}{4} x^2 + y^2 = 1 \), oriented counter-clockwise.

Key Concepts

line integrals

Line integrals play a crucial role in mathematics, especially in fields like physics and engineering. Simply put, a line integral is a type of integral where function values are summed along a curve or path.
Instead of integrating over an interval as you do for a typical integral, you integrate over a path. This is extremely useful for computing things like work done by a force within a vector field. For example, if a force is applied in a vector field, the line integral helps calculate the work done by that force along a specific path.

• The function is integrated along a curve or path.
• Commonly used for calculating work done by fields.
• Helps transform problems into more manageable forms.

In practical terms, if you had a path defined by a parameterization, say from point A to point B, the line integral sums up all the infinitesimally small work portions as you move from A to B. This works well with Green's Theorem, which allows you to convert these line integrals into easier double integrals.

vector fields

A vector field is essentially a function that assigns a vector to every point within a certain space. In the case of a plane vector field, a vector is assigned to every point in the plane.
Think of it like a map where at each point, instead of just having coordinates, you have a vector showing direction and magnitude. Bishop to the northwest with a vector arrow instead of just saying "point B."

• Assigns vectors throughout a space.
• Visualizes directions and magnitudes at all points.
• Used to model diverse phenomena, like flows or fields.

In practice, vector fields are widely used to model physical phenomena such as magnetic fields or fluid flows. Each vector in the field can represent, for example, the speed and direction of wind at each point on a map.

curl of a vector field

The curl of a vector field is a measure of how much and in which direction a vector field 'rotates' around a point. It’s like understanding how a field 'spins' around an axis. This is particularly useful when you are dealing with vector fields or flows.
Mathematically, the curl is a vector itself, derived from a derivative operation, and is given by the expression:
\[ abla \times \mathbf{F} \] where \( \mathbf{F} \) is your vector field.

• Measures the rotational effect in a vector field.
• The result can be deduced using partial derivatives.
• Helps visualize circulation within fields.

For example, if you're analyzing a fluid flow, the curl could show areas where there is likely to be vortex formation. The concept is also pivotal when applying Green's Theorem, converting a line integral to a double integral by taking the curl of a vector field.

double integrals

Double integrals extend the concept of integrals to functions of two variables, allowing you to compute the accumulation of quantities over a region in a plane. Imagine you want to find the total mass of a thin sheet with varying density. Double integrals come into play here.
To perform a double integral, you integrate a function first with respect to one variable, and then integrate that result with respect to the second variable, typically convering an area described by two variables.

• Used for calculating areas and accumulations like mass.
• Integrates functions over two-dimensional regions.
• Key tool for solving area integrals in applied mathematics.

Double integrals are particularly useful in applying the Green's Theorem, as they allow a transformation of line integrals around closed curves into more manageable double integrals over the region bounded by those curves.