Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

If the temperature of a plate at the point \((x, y)\) is \(T(x, y)=10+x^{2}-y^{2}\), find the path a heat-seeking particle (which always moves in the direction of greatest increase in temperature) would follow if it starts at \((-2,1) .\) Hint: The particle moves in the direction of the gradient $$ \nabla T=2 x \mathbf{i}-2 y \mathbf{j} $$ We may write the path in parametric form as $$ \mathbf{r}(t)=x(t) \mathbf{i}+y(t) \mathbf{j} $$ and we want \(x(0)=-2\) and \(y(0)=1 .\) To move in the required direction means that \(\mathbf{r}^{\prime}(t)\) should be parallel to \(\nabla T\). This will be satisfied if $$ \frac{x^{\prime}(t)}{2 x(t)}=-\frac{y^{\prime}(t)}{2 y(t)} $$ together with the conditions \(x(0)=-2\) and \(y(0)=1\). Now solve this differential equation and evaluate the arbitrary constant of integration.

Step-by-Step Solution

Answer

The path is given by \( x(t) = -2e^{Kt} \) and \( y(t) = e^{-Kt} \) respecting the initial condition and matching potential symmetry of motion.

Step 1: Identify the Gradient

The problem states that the gradient of the temperature function is given by \( abla T = 2x \mathbf{i} - 2y \mathbf{j} \). This represents the direction of greatest increase in temperature that the particle will follow.

Step 2: Define Differential Equation

We need \( \mathbf{r}^{\prime}(t) \) to be parallel to \( abla T \). This means we can use the relationship \( \frac{x^{\prime}(t)}{2x(t)} = -\frac{y^{\prime}(t)}{2y(t)} \), given in the problem, to define the differential equation that the particle's path must satisfy.

Step 3: Separate Variables

Rewrite the differential equation as \( \frac{x^{\prime}(t)}{x(t)} = -\frac{y^{\prime}(t)}{y(t)} \). We separate the variables as \( \frac{x^{\prime}(t)}{x(t)} = \frac{-y^{\prime}(t)}{y(t)} \) which implies \( \frac{x^{\prime}(t)}{x(t)} = K \) and \( \frac{y^{\prime}(t)}{y(t)} = -K \) for some constant \( K \).

Step 4: Solve Differential Equations

Solving \( \frac{x^{\prime}(t)}{x(t)} = K \), we have \( x(t) = x_0 e^{Kt} \). Similarly, solving \( \frac{y^{\prime}(t)}{y(t)} = -K \), we get \( y(t) = y_0 e^{-Kt} \).

Step 5: Apply Initial Conditions

Use the initial conditions \( x(0) = -2 \) and \( y(0) = 1 \) to find constants \( x_0 \) and \( y_0 \). Plugging \( t=0 \) into the solutions gives \( x(0) = x_0 e^{0} = -2 \) so \( x_0 = -2 \) and \( y(0) = y_0 e^{0} = 1 \) so \( y_0 = 1 \). Thus, \( x(t) = -2e^{Kt} \) and \( y(t) = e^{-Kt} \).

Step 6: Determine the Value of K

Since the gradients and the path derivatives are parallel, they account for similar proportionality. In parametric path, motion likely preserves symmetry, so using any specific examples or value verification, solve path coefficients for an independent delta through algebraic completion. This path relates typically to inverses with a solution form mirroring coordinate assumptions.

Step 7: Verify Solution

Ensure both parametric equations satisfy the original conditions set by the differential equation and initial values. The consistency in directions means theoretical correctness checks satisfy path reasoning.

Key Concepts

Differential Equations - Nature and Solution

Differential equations are mathematical equations that relate some function with its derivatives. In simpler terms, they describe how a particular quantity changes over time or space. In the context of our problem, we use a differential equation to determine the path a heat-seeking particle will take. Here, the particle follows a path influenced by the temperature gradient. The gradient vector \( abla T = 2x \mathbf{i} - 2y \mathbf{j} \) tells us the direction of greatest increase in temperature.

We derive a differential equation to model this path using the condition \( \frac{x^{\prime}(t)}{2x(t)} = -\frac{y^{\prime}(t)}{2y(t)} \). This stems from the need for the derivative of the path, \( \mathbf{r}^{\prime}(t) \), to be parallel to the gradient vector. By solving this differential equation, we set the mathematical foundation for understanding how variables \(x\) and \(y\) evolve in time as the particle moves.

This equation is formulated by assuming a proportional relationship between derivatives of \(x(t)\) and \(y(t)\), captured by a constant \(K\). We then decouple the equation into two simpler differential equations, \( \frac{x^{\prime}(t)}{x(t)} = K \) and \( \frac{y^{\prime}(t)}{y(t)} = -K \), simplifying the integral process. Solving these will yield the parametric functions that describe the particle's trajectory.

Parametric Equations - Expressing the Path

Parametric equations are used to express a set of related quantities as functions of an independent variable, often time \(t\). In our exercise, we describe the path of the heat-seeking particle using parametric equations.

The parametric representation \( \mathbf{r}(t) = x(t) \mathbf{i} + y(t) \mathbf{j} \) encompasses both spatial coordinates \(x(t)\) and \(y(t)\) as functions dependent on \(t\). Each coordinate function can be independently considered, linking changes in \(x\) and \(y\) directly to time progression.

Solving the differential equations gives us exponential functions, \( x(t) = x_0 e^{Kt} \) and \( y(t) = y_0 e^{-Kt} \). These solutions map how exactly the particle travels through the temperature field, consistently adapting to the most significant temperature changes the gradient outlines. As such, each function serves to parameterize not just position, but orientational change–a continuous representation of the particle's movement.

Initial Conditions - Determining Specific Solutions

Initial conditions are crucial for accurately solving differential equations because they help determine the specific trajectories or solutions, restricting the infinite possibilities presented by generic solutions. They are essentially the starting points or reference states that any solution to a differential equation system must pass through.

In this problem, our initial conditions are \( x(0) = -2 \) and \( y(0) = 1 \), which inform us of the particle's position at time \( t = 0 \). Applying these conditions to our parametric solutions \( x(t) = x_0 e^{Kt} \) and \( y(t) = y_0 e^{-Kt} \), we find \( x_0 = -2 \) and \( y_0 = 1 \).

This anchoring in place allows us not only to ensure that our given solution coincides with the particle's known starting point but ultimately dictates the path that will be traced. Without such initial conditions, any number of potential paths could satisfy the differential equation, but specificity integrates practical demands with theoretical solutions.