Problem 44 - Calculus Early Transcendentals

Question

Find the work done by the force field \(\mathbf{F}\) on a particle that moves along the curve \(C\). \(\mathbf{F}(x, y, z)=(x+y) \mathbf{i}+x y \mathbf{j}-z^{2} \mathbf{k}\) \(C:\) along line segments from \((0,0,0)\) to \((1,3,1)\) to \((2,-1,4)\)

Step-by-Step Solution

Answer

The total work is found by summing integrals over both segments.

Step 1: Understand the Line Integral for Work

The work done by a force field \(\mathbf{F}\) along a curve \(C\) is calculated using the line integral \(W = \int_C \mathbf{F} \cdot d\mathbf{r}\). Here, \(d\mathbf{r}\) represents a differential element of the path.

Step 2: Break Curve into Segments

The curve \(C\) consists of two straight line segments: from \((0,0,0)\) to \((1,3,1)\), and from \((1,3,1)\) to \((2,-1,4)\). We need to parametrize each segment separately and compute the integral for each.

Step 3: Parametrize the First Segment

For the first segment from \((0,0,0)\) to \((1,3,1)\), we use the parameter \(0 \le t \le 1\) as \(\mathbf{r}_1(t) = t\mathbf{i} + 3t\mathbf{j} + t\mathbf{k}\).

Step 4: Compute \(d\mathbf{r}_1\) and Substitute into Work Integral

The differential \(d\mathbf{r}_1 = (1\mathbf{i} + 3\mathbf{j} + 1\mathbf{k}) dt\). Substitute \(\mathbf{r}_1(t)\) into \(\mathbf{F}(x,y,z)\) to get \(\mathbf{F} = (4t)\mathbf{i} + (3t^2)\mathbf{j} - t^2\mathbf{k}\). Then, compute the dot product: \(\mathbf{F} \cdot d\mathbf{r}_1 = (4t)(1) + (3t^2)(3) + (-t^2)(1) dt\).

Step 5: Evaluate the First Integral

Evaluate \(\int_0^1 (4t + 9t^2 - t^2) dt = \int_0^1 (4t + 8t^2) dt\). Integrate to find \([2t^2 + \frac{8}{3}t^3]_0^1\), which results in \(\frac{14}{3}\).

Step 6: Parametrize the Second Segment

For the second segment from \((1,3,1)\) to \((2,-1,4)\), use \(0 \le s \le 1\) with \(\mathbf{r}_2(s) = (1+s)\mathbf{i} + (3-4s)\mathbf{j} + (1+3s)\mathbf{k}\).

Step 7: Compute \(d\mathbf{r}_2\) and Substitute into Work Integral

The differential \(d\mathbf{r}_2 = (1\mathbf{i} - 4\mathbf{j} + 3\mathbf{k}) ds\). Substitute \(\mathbf{r}_2(s)\) into \(\mathbf{F}(x,y,z)\) to get \(\mathbf{F} = (4 + s - 4s)\mathbf{i} + (3 - 4s)(1 + s)\mathbf{j} - (1 + 3s)^2\mathbf{k}\). Evaluate \(\mathbf{F} \cdot d\mathbf{r}_2\).

Step 8: Evaluate the Second Integral

Compute \(\int_0^1 (4 - 3s) + (3 - 3s)s)(-7 - 6s) ds\) to find the contribution of the second segment.

Step 9: Sum the Work from Both Segments

Add the results from both integrals: \(\frac{14}{3}\) from the first segment and the integral for the second segment to find the total work.

Key Concepts

Line Integrals

Line integrals are a fundamental concept in calculus used to calculate the accumulation of values along a curve. Imagine walking along a path and measuring a variable such as temperature, elevation, or in our case, the influence of a force field. Line integrals help calculate the total effect over that path.

• For work, the line integral measures the accumulated effort exerted by a vector field, such as a force, as an object moves along a particular path.
• This requires evaluating the integral of the dot product of a force vector and the differential path element (often written as \(d\mathbf{r}\)).
• The formula is generally given as \(W = \int_C \mathbf{F} \cdot d\mathbf{r}\), where \(\mathbf{F}\) is the vector field and \(C\) represents the path or curve.

To solve such problems, one typically represents the path using parametric equations, which simplifies converting the integral along the path. By transforming the curve into simpler mathematical expressions, computing the total work along the curve becomes manageable.

Work in Force Fields

Work in force fields is a concept that helps us understand how force influences an object as it moves through space. Think of it as how much energy a field, like gravity or magnetism, expends on an object to move it.

• In vector calculus, force fields are often represented as vector fields, meaning they assign a vector to every point in space, indicating the direction and magnitude of the force at that point.
• Calculating work done in a force field involves evaluating how the field affects the object along its path from an initial to a final position.
• This is accomplished using the concept of line integrals, to assess the effect of the field along a curve.

A key step is parameterizing the path into segments when it's piecewise linear, as objects might traverse through multiple regions of influence. In every portion of the path, understanding and computing the contribution of the force, before integrating along the entire path is essential, as demonstrated in the line integral solution process above.

Vector Calculus

Vector calculus is a branch of mathematics focused on vector fields and their derivatives and integrals. These concepts are essential in defining and understanding multi-dimensional variables and their behaviors.

• Vectors represent quantities with both magnitude and direction, such as force, velocity, or acceleration.
• Vector calculus techniques, like line integrals, surface integrals, and divergence, help describe how these vectors change and influence their environment.
• In our exercise, vector calculus allows the calculation of work done by a force field on a particle moving through space.

Vector calculus brings together algebra and calculus, enabling us to solve complex physical problems involving force fields and motion. By parameterizing curves and using derivatives and integrals, it allows us to model real-world phenomena effectively, showing how vectors operate within different reference frames and constraints.