Problem 42 - Advanced Engineering Mathematics

Question

The temperature \(T\) at a point \((x, y, z)\) in space is inversely proportional to the square of the distance from \((x, y, z)\) to the origin. It is known that \(T(0,0,1)=500\). Find the rate of change of \(T\) at \((2,3,3)\) in the direction of \((3,1,1)\). In which direction from \((2,3,3)\) does the temperature \(T\) increase most rapidly? At \((2,3,3)\) what is the maximum rate of change of \(T ?\)

Step-by-Step Solution

Answer

The maximum rate of temperature increase at (2,3,3) is the magnitude of the gradient. Temperature increases most rapidly in the direction of the gradient.

Step 1: Understanding the Inverse Proportionality

The temperature at \(T(x, y, z)\) is inversely proportional to the square of the distance from the origin. This relationship is given by \(T(x, y, z) = \frac{C}{x^2 + y^2 + z^2}\), where \(C\) is a constant.

Step 2: Determining the Constant

Use the given condition \(T(0,0,1) = 500\). The distance is 1, so \(T(0, 0, 1) = \frac{C}{1^2} = C = 500\). Thus, \(T(x, y, z) = \frac{500}{x^2 + y^2 + z^2}\).

Step 3: Compute the Gradient of T

The gradient of \(T\) is \(-abla T = \left(-\frac{\partial T}{\partial x}, -\frac{\partial T}{\partial y}, -\frac{\partial T}{\partial z}\right)\).\ Compute each component: \(\frac{\partial T}{\partial x} = -\frac{1000x}{(x^2 + y^2 + z^2)^2}\),\(\frac{\partial T}{\partial y} = -\frac{1000y}{(x^2 + y^2 + z^2)^2}\),\(\frac{\partial T}{\partial z} = -\frac{1000z}{(x^2 + y^2 + z^2)^2}\).

Step 4: Evaluate the Gradient at (2, 3, 3)

Substitute \(x = 2, y = 3, z = 3\) into the gradient formula:\(abla T(2, 3, 3) = \left(-\frac{1000 \cdot 2}{(2^2 + 3^2 + 3^2)^2}, -\frac{1000 \cdot 3}{(2^2 + 3^2 + 3^2)^2}, -\frac{1000 \cdot 3}{(2^2 + 3^2 + 3^2)^2}\right)\)After calculation, \(abla T(2, 3, 3)\) will be a specific vector based on these values.

Step 5: Determine the Directional Derivative

To find the rate of change of \(T\) in the direction of \(\mathbf{v} = (3, 1, 1)\), normalize \(\mathbf{v}\) to get its unit vector \(\mathbf{u}\). \(\mathbf{u} = \frac{1}{\sqrt{3^2 + 1^2 + 1^2}} (3, 1, 1)\).The directional derivative is given by \(abla T \cdot \mathbf{u}\).

Step 6: Calculate the Maximum Rate of Temperature Increase

The maximum rate of increase of temperature is given by the magnitude of the gradient, \(| abla T(2, 3, 3) |\). Compute this using the values found in Step 4.

Step 7: Identify the Direction for Maximum Increase

The temperature increases most rapidly in the direction of the gradient vector \(abla T(2, 3, 3)\). This vector itself points in the direction of the steepest ascent of \(T\).

Key Concepts

Directional Derivative

The directional derivative is a concept in vector calculus that helps us understand how a function changes as we move in a specific direction. It's like asking, "If I go this way, how quickly will the temperature change?" In the context of our problem, the temperature function at a point in space can change at different rates, depending on which direction you move.
To calculate the directional derivative, you first need a direction. This direction is represented as a vector, often denoted as \(\mathbf{v}\). Then, you must convert \(\mathbf{v}\) to a unit vector \(\mathbf{u}\) to ensure it has a magnitude of 1. Units matter because the directional derivative tells us how much the function is changing per unit length in that direction.

• Find the unit vector by dividing each component of \(\mathbf{v}\) by the vector's magnitude.
• Compute the directional derivative by taking the dot product of the gradient of the function with this unit vector. If the gradient vector at a point is \(abla T\) and your unit vector is \(\mathbf{u}\), then the directional derivative is \(abla T \cdot \mathbf{u}\).

This value gives you the rate of change of the function, like temperature, at the point, as you move in the direction of \(\mathbf{v}\).

Gradient Vector

The gradient vector is a powerful tool in vector calculus that displays the direction of steepest ascent for a function. Think of it as a navigator, always pointing towards the direction in which the value of the function, such as temperature, increases the fastest. For our temperature function, the gradient vector is written as \(abla T = \left(\frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}, \frac{\partial T}{\partial z}\right)\).
This vector not only identifies direction but also indicates magnitude. The size of the gradient vector at a certain point tells us the maximum rate of increase of the function at that point.

• The gradient vector shows where to move in 3D space so that the rate of increase of the function is maximal.
• Its components, partial derivatives, determine how the function changes as each variable changes independently.

In the exercise, calculating the gradient at \((2, 3, 3)\) involves substituting these coordinates into the partial derivative formulas derived earlier. The resulting vector then shows both the direction and the steepness of the temperature increase at that point.

Inverse Proportionality

Inverse proportionality is a simple yet significant concept telling us that one quantity decreases as another increases. In our exercise, the temperature \(T\) at a point in space becomes smaller as the distance from the origin grows. Mathematically, this relationship is articulated as \(T(x, y, z) = \frac{C}{x^2 + y^2 + z^2}\), where \(C\) is a constant.
Here, the distance from the point \((x, y, z)\) to the origin is symbolized by the denominator \(x^2 + y^2 + z^2\). This formula suggests that:

• When the point \((x, y, z)\) is close to the origin, the temperature \(T\) is high because the denominator is small.
• Conversely, as \((x, y, z)\) moves further away, the temperature decreases since the denominator grows larger.

In the exercise, this principle helps determine the constant required to describe at which the rate changes, especially when given specific values like \(T(0,0,1) = 500\). This relationship is crucial for analyzing directional derivatives and gradient vectors in context.

Rate of Change

The rate of change is a concept that dictates how quickly a particular variable alters concerning another. In an intuitive manner, it's analogous to how swiftly the temperature might change as you wander in a particular direction or change your position in space. It's vital to understand it in connection with the directional derivative and gradient.
The rate of change can be seen as:

• The magnitude of the gradient vector providing the maximum rate of temperature change.
• The directional derivative shows the rate of change in a specific direction.

A higher rate might mean a rapid increase in temperature as you step in a certain direction or position. Conversely, a lower rate signifies minimal change. Calculated via these methods, it helps understand and predict the behavior of the function in concrete terms, aligning all three concepts to make sense of how a vector calculus problem unfolds.