Problem 47 - Thomas Calculus

Question

\(\mathbf{F}\) is the velocity field of a fluid flowing through a region in space. Find the flow along the given curve in the direction of increasing \(t .\) $$ \begin{array}{l}{\mathbf{F}=-4 x y \mathbf{i}+8 y \mathbf{j}+2 \mathbf{k}} \\\ {\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 2}\end{array} $$

Step-by-Step Solution

Answer

The flow along the curve is 48.

Step 1: Clarify the Problem

We need to calculate the flow along a curve defined by a vector function \(\mathbf{r}(t)\) within a given velocity field \(\mathbf{F}\). This involves computing the line integral of \(\mathbf{F}\) over the curve \(\mathbf{r}(t)\) from \(t = 0\) to \(t = 2\).

Step 2: Compute the Derivative

First, find the derivative of the parameterized curve \(\mathbf{r}(t) = t\mathbf{i} + t^2\mathbf{j} + \mathbf{k}\), which gives \(\mathbf{r}'(t) = \frac{d\mathbf{r}}{dt} = \mathbf{i} + 2t\mathbf{j}\).

Step 3: Evaluate the Field on the Curve

Substitute \(x = t\), \(y = t^2\), and \(z = 1\) into \(\mathbf{F}\) because \(\mathbf{r}(t) = t\mathbf{i} + t^2\mathbf{j} + \mathbf{k}\). Thus, \(\mathbf{F}(t) = -4t(t^2)\mathbf{i} + 8(t^2)\mathbf{j} + 2\mathbf{k} = -4t^3\mathbf{i} + 8t^2\mathbf{j} + 2\mathbf{k}\).

Step 4: Set Up the Line Integral

The line integral of \(\mathbf{F}\) along \(\mathbf{r}(t)\) can be written as \(\int_{a}^{b} \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) \, dt\), where \(a = 0\) and \(b = 2\).

Step 5: Compute the Dot Product

Compute the dot product: \((-4t^3\mathbf{i} + 8t^2\mathbf{j} + 2\mathbf{k}) \cdot (\mathbf{i} + 2t\mathbf{j})\). This gives \(-4t^3 + 16t^3 = 12t^3\). Since the \(\mathbf{k}\) components disappear from dot product computation, the result is \(12t^3\).

Step 6: Evaluate the Integral

The integral is \(\int_0^2 12t^3 \, dt\). To solve it, integrate: \(\int 12t^3 \, dt = 12 \cdot \frac{t^4}{4} = 3t^4\). Evaluate it from 0 to 2: \([3(2^4) - 3(0^4)] = [3(16) - 0] = 48\).

Step 7: State the Result

The flow along the curve in the direction of increasing \(t\) is 48.

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields, which are functions that assign a vector to every point in space. This area of calculus is crucial for understanding how vectors behave in different regions and how they interact with surfaces and paths.
One of the main tools in vector calculus is the line integral. This is used to evaluate the behavior of a vector field along a path or curve. In simpler terms, it's like measuring how a force field acts on a moving object along its path. The line integral calculates the cumulative effect or influence of the vector field on the object, typically by integrating along a curve or path in the field.
In our exercise, we calculate the flow of a fluid through a vector field along a specified curve. Line integrals are particularly useful in physics and engineering for applications like electromagnetism and fluid dynamics, where they help illustrate how forces like magnetic or fluid flow fields act over a given trajectory.

Velocity Field

A velocity field is a vector field that represents the velocity of a fluid at every point in space. This is fundamental in fluid dynamics, as it helps describe how the fluid moves throughout a region. Each vector in the field shows the direction and speed of the fluid at that specific point.
In the given problem, the velocity field \( \mathbf{F} = -4xy \mathbf{i} + 8y \mathbf{j} + 2 \mathbf{k} \) assigns a velocity vector to each point in a 3-dimensional space. Understanding this can help us predict how an object will experience forces or "flow" when moving along a specific path within the field.
Analyzing this vector field is like looking at a wind map: The arrows (vectors) demonstrate how strong the wind is and which direction it is blowing at different locations. This information can be crucial in predicting the behavior of objects moving within that field, whether it's an airplane in the sky or a pebble in a stream.

Parametrized Curves

Parametrized curves involve expressing a curve in terms of a parameter, usually denoted as \( t \). This representation allows for easier computation and description of more complex curves, especially in three dimensions.
In vector calculus, a parametrized curve, like the one given by \( \mathbf{r}(t) = t\mathbf{i} + t^2\mathbf{j} + \mathbf{k} \, \ 0 \leq t \leq 2 \), allows us to compute line integrals by providing an explicit form of the curve that a line integral can be applied to. The parameter \( t \) varies over an interval, giving us a way to trace the curve in space.
In practical terms, you can think of it like plotting a path on a map where \( t \) is time. As \( t \) changes, the curve denotes a trajectory through space. This is helpful for visualizing and calculating the interaction of curves with vector fields, as it allows for a step-by-step construction of the path and makes complex integrations manageable.