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.

Short answer

The direction of greatest increase in temperature is toward or away from the origin.

Step-by-step solution

Define the Temperature Function

Since the temperature depends only on the distance from the origin, we express the temperature as a function of the distance: let \( T = f(r) \), where \( r = \sqrt{x^2 + y^2 + z^2} \). The function \( f \) describes how the temperature changes with respect to the distance \( r \).

Express the Gradient of the Temperature Function

The gradient of the temperature \( abla T \) is the vector that points in the direction of greatest increase of \( T \). Using the chain rule, \( abla T = f'(r) abla r \). The gradient \( abla r = \left(\frac{x}{r}, \frac{y}{r}, \frac{z}{r}\right) \).

Simplify the Gradient Expression

Multiply the expression for \( abla r \) by \( f'(r) \), resulting in \( abla T = f'(r) \left(\frac{x}{r}, \frac{y}{r}, \frac{z}{r}\right) = \frac{f'(r)}{r} \left(x, y, z\right) \).

Interpret the Direction of the Gradient Vector

The gradient \( abla T \) points in the direction \( \left(x, y, z\right) \). Since \( x, y, \) and \( z \) are the coordinates of the point \( (x, y, z) \), the vector \( \left(x, y, z\right) \) either points towards the origin (if \( f'(r) < 0 \)) or away from the origin (if \( f'(r) > 0 \)).

Key concepts

Gradient Vector

The gradient vector, denoted as \( abla T \), is crucial in multivariable calculus. It indicates the direction of the maximum rate of increase of a scalar field, like temperature. In simpler terms, if you were standing on a hill and wanted to find the direction that steepest ascends, the gradient would lead you there. The gradient not only shows us the direction but its magnitude gives us the steepness of the increase.
The formula for the gradient vector in a three-dimensional space such as \((x, y, z)\) is the partial derivatives of the function concerning each variable. In this exercise, however, because the temperature depends only on the distance from the origin, the gradient simplifies to incorporate the derivative of the function with respect to \( r \) and the directional components \((x, y, z)\).
In practical scenarios, the gradient vector gets expressed as a combination of these coordinate directions scaled by the rate of change at \( r \). This helps us visualize how changes occur with respect to distance changes, rather than individual coordinate axis changes.

Chain Rule

The chain rule is a fundamental tool in calculus for dealing with composite functions. It allows us to differentiate functions that have other functions inside of them. For example, in the context of the temperature function \( T = f(r) \), we observe that temperature \( T \) changes concerning the distance \( r \), itself a function of \((x, y, z)\).
By applying the chain rule, we can find how changes in \((x, y, z)\) impact the temperature \( T \). Specifically, the chain rule formula in this context helps us express \( abla T = f'(r) abla r \). This means we multiply the derivative of the temperature function concerning \( r \) with the gradient of \( r \).
This technique is key when working with functions that are layered, allowing us to "chain" or link the derivatives together and compute them accurately.

• Understand relationships between variables.
• Calculate derivative for composite or nested functions.

Directional Derivatives

Directional derivatives extend the concept of a derivative to measure how a function changes as we move in any particular direction. In three dimensions, rather than limiting ourselves to the x, y, or z directions individually, we can observe changes in any direction we choose.
For temperature depending on the distance from the origin, the directional derivative tells us how much the temperature changes not just at a position, but specifically as we move either towards or away from the origin. Here, the directional derivative can be represented by taking the gradient of the temperature \( abla T \) and finding its component in the direction of movement.
This component is crucial in understanding how small alterations in direction can affect rates of change in different paths, helping us map and predict changes in fields like temperature based on direction.

• Analyze function behavior in any given direction.
• Link derivative concepts with spatial directions.

Distance from the Origin

The distance from the origin (0,0,0) in a three-dimensional space can be calculated using the formula \( r = \sqrt{x^2 + y^2 + z^2} \). This measure is fundamental because it gives us a scalar value that represents how far away any point \((x, y, z)\) is from the central point of reference.
In this exercise, understanding how temperature \( T \) changes with respect to this distance \( r \) is crucial. We express the temperature not in terms of each coordinate individually but as a function of the distance itself \( T = f(r) \).
This approach is beneficial when dealing with spheres or symmetrical objects where changes purely depend on distance, allowing for easier analysis and derivation of changes in temperature or other scalar fields.

• Calculates how far a point is from the origin.
• Simplifies understanding of spherical symmetry.