Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that the temperature \(T\) at the point \((x, y, z)\) depends only on the distance from the origin. Show that the direction of greatest increase in \(T\) is either directly toward the origin or directly away from the origin.

Step-by-Step Solution

Answer

The direction of greatest increase in temperature is either toward or away from the origin, depending on the sign of \( \frac{df}{dr} \).

Step 1: Define the Temperature Function

Since the temperature \( T \) depends only on the distance from the origin, let's express it as a function of the distance \( r \), where \( r = \sqrt{x^2 + y^2 + z^2} \). We can write \( T = f(r) \), meaning that the temperature is a function of \( r \).

Step 2: Express in Gradient Terms

The gradient of a function \( T(x, y, z) \) gives the direction of the greatest increase. The gradient is expressed as \( abla T = \left( \frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}, \frac{\partial T}{\partial z} \right) \).

Step 3: Chain Rule for Partial Derivatives

Since \( T = f(r) \) where \( r = \sqrt{x^2 + y^2 + z^2} \), apply the chain rule. Compute \( \frac{\partial T}{\partial x} = \frac{df}{dr} \cdot \frac{\partial r}{\partial x} \), and similarly for \( y \) and \( z \).

Step 4: Calculate Partial Derivatives of r

The partial derivatives of \( r \) are \( \frac{\partial r}{\partial x} = \frac{x}{r} \), \( \frac{\partial r}{\partial y} = \frac{y}{r} \), \( \frac{\partial r}{\partial z} = \frac{z}{r} \).

Step 5: Formulate Gradient of T

Substitute these into the chain rule expression: \(abla T = \frac{df}{dr} \left( \frac{x}{r}, \frac{y}{r}, \frac{z}{r} \right) = \frac{df}{dr} \hat{r} \), where \( \hat{r} = \left( \frac{x}{r}, \frac{y}{r}, \frac{z}{r} \right) \) is the unit vector in the direction of \( \vec{r} \).

Step 6: Analyze the Direction of the Gradient

The gradient \( abla T = \frac{df}{dr} \hat{r} \) indicates the direction of greatest change. If \( \frac{df}{dr} > 0 \), the direction of greatest increase in \( T \) is the same as \( \hat{r} \) (i.e., away from the origin). If \( \frac{df}{dr} < 0 \), the direction of increase is the opposite of \( \hat{r} \) (i.e., toward the origin). Thus, the direction of greatest increase in \( T \) is either directly toward or away from the origin, depending on the sign of \( \frac{df}{dr} \).

Key Concepts

Gradient

The concept of the gradient is central in multivariable calculus, as it helps determine the direction of greatest increase of a scalar field, like temperature. The gradient of a function is represented as a vector made up of partial derivatives, providing both direction and rate of change in a multidimensional space.
For a function of several variables, say, temperature \( T(x, y, z) \), the gradient \( abla T \) is given by:

• \( abla T = \left( \frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}, \frac{\partial T}{\partial z} \right) \)

This vector points in the direction where the function increases most rapidly, with its magnitude indicating the rate of increase. In our exercise, since temperature \( T \) depends only on distance \( r \) from the origin, the gradient simplifies to a direction toward or away from the origin, depending on the derivative of \( T \) with respect to \( r \).
Understanding how to navigate these gradients allows us to predict and describe changes within spatial fields, making it essential in physics and engineering.

Directional Derivative

The directional derivative extends the concept of a gradient by specifying a particular direction for the rate of change. While a gradient finds the direction of the steepest ascent, the directional derivative measures how a function changes as we move in any specified direction, meaning it can be applied in diverse situations.
To compute the directional derivative of \( T \) at point \((x, y, z)\) in the direction of a unit vector \( \mathbf{u} = (u_x, u_y, u_z)\), the formula is:

• \( D_u T = abla T \cdot \mathbf{u} = \frac{\partial T}{\partial x} u_x + \frac{\partial T}{\partial y} u_y + \frac{\partial T}{\partial z} u_z \)

This measures how \( T \) changes as we move at an angle, specifically in the direction provided by \( \mathbf{u} \), capturing changes not just limited to the gradient's path. In our case, the interest lies whether this directional derivative increases or decreases towards or away from the origin.

Partial Derivatives

Partial derivatives are the building blocks of the gradient and are crucial for understanding changes in multivariable functions. They represent the rate of change of a function with respect to one of its variables, while keeping the other variables constant.
For a function \( T(x, y, z) \), the partial derivatives are:

• \( \frac{\partial T}{\partial x} \)
• \( \frac{\partial T}{\partial y} \)
• \( \frac{\partial T}{\partial z} \)

In the context of our exercise, where \( T \) depends on the distance \( r \), these partial derivatives are handled using the chain rule, converting them into derivatives dependent on \( r \). This transformation helps express the gradient in terms of distance from the origin, connecting the partial derivatives to the physical concept of temperature change in space.

Vector Calculus

Vector calculus involves understanding and working with functions that have directional attributes and are represented as vectors, crucial for describing physical phenomena in physics and engineering. When dealing with functions like our temperature \( T(x, y, z) \), vector calculus helps us describe the function's behavior as it varies in space.
Key operations in vector calculus include gradients, divergences, and curls. For the exercise provided, the focus is on evaluating the gradient, a foundational vector calculus operation that provides insights into how scalar fields like temperature vary spatially.

• Gradient: Tells us directions and rates of change.
• Position Vectors: Communicate locations in space, associated with each point \((x, y, z)\).

Through vector calculus, we analyze the underlying vector fields and ascertain directions of maximal change, turning abstract mathematical concepts into tangible, real-world interpretations, such as comprehending how temperature rises or falls in different parts of a given space.