Question

Zeige, daß \(\varphi(\underline{x})=x_{1}^{m}+x_{2}^{m}+\ldots+x_{n}^{m} \quad(m \in \mathbb{N})\) die Gleichung \(\underline{x} \cdot \operatorname{grad} \varphi(\underline{x})=m \varphi(\underline{x})\) erfüllt.

Short answer

The function \( \varphi(\underline{x})=x_{1}^{m}+x_{2}^{m}+\ldots+x_{n}^{m} \) satisfies the equation \( \underline{x} \cdot \operatorname{grad} \varphi(\underline{x})=m \varphi(\underline{x}) \) for all \( \underline{x} \). This has been proved by calculating both the gradient of \( \varphi \) and the dot product between \( \underline{x} \) and \( \operatorname{grad} \varphi \), and showing that they indeed produce the same results.

Step-by-step solution

Compute the gradient

The gradient is a vector field, and it is computed by taking the derivative of \( \varphi \) with respect to each dimension, resulting in a vector of these partial derivatives. Thus, the gradient of \( \varphi \) is \( \operatorname{grad} \varphi = (\frac{\partial \varphi}{\partial x_1}, \frac{\partial \varphi}{\partial x_2}, ..., \frac{\partial \varphi}{\partial x_n}) \). For \( \varphi(\underline{x})=x_{1}^{m}+x_{2}^{m}+\ldots+x_{n}^{m} \), the partial derivative with respect to \( x_i \) is \( m x_{i}^{m-1} \). Therefore, \( \operatorname{grad} \varphi = (m x_{1}^{m-1}, m x_{2}^{m-1}, ..., m x_{n}^{m-1}) \).

Compute the dot product

The dot product between \( \underline{x} \) and \( \operatorname{grad} \varphi \) is given by \( \underline{x} \cdot \operatorname{grad} \varphi = x_{1} \cdot m x_{1}^{m-1} + x_{2} \cdot m x_{2}^{m-1} + ... + x_{n} \cdot m x_{n}^{m-1} = m x_{1}^{m} + m x_{2}^{m} + ... + m x_{n}^{m} = m \varphi(\underline{x}) \).

Conclusion

From the previous step, the computed value of \( \underline{x} \cdot \operatorname{grad} \varphi \) is indeed \( m \varphi(\underline{x}) \) for all \( \underline{x} \). Thus, it is proved that \( \underline{x} \cdot \operatorname{grad} \varphi(\underline{x})=m \varphi(\underline{x}) \), as required.

Key concepts

Vector Analysis

Vector analysis is a pivotal field in mathematics that deals with operators on vector fields and functions. It includes operations like the gradient, divergence, and curl, which are essential in physics and engineering to describe fields like electromagnetism and fluid dynamics.
One integral operation in vector analysis is the computation of the gradient. The gradient of a scalar function indicates the direction and rate of the fastest increase of the function. In the context of our exercise, where the function \( \varphi(\underline{x})=x_{1}^{m}+x_{2}^{m}+\ldots+x_{n}^{m} \), the gradient is a vector of the partial derivatives with respect to each variable. It's the generalization of the derivative concept from calculus, extending it to multiple dimensions.
In practical terms, finding a gradient involves calculating how much the function changes when each variable is tweaked slightly—a concept that is useful in optimization, where one might want to find the highest or lowest point on a surface.

Partial Derivative

The partial derivative represents the rate at which a function changes with respect to one of its variables, while holding the other variables constant. It's a fundamental concept in multivariable calculus and is denoted as \( \frac{\partial}{\partial x_i} \) for a variable \( x_i \).
In our exercise, the function \( \varphi(\underline{x}) \) has to be differentiated partially with respect to each variable \( x_i \), leading to \( m x_{i}^{m-1} \). This step shows how each component of the input vector contributes to the function's rate of change individually. Understanding partial derivatives is essential for gradient computation because the gradient itself is a collection of partial derivatives—a vector pointing in the direction of the steepest ascent.
In real-world applications, partial derivatives are used to analyze systems with multiple variables, such as in economic models to determine how different factors affect outcomes, or in thermodynamics to study how properties like pressure and temperature change within a system.

Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation is denoted as \( \underline{a} \cdot \underline{b} \) and can be geometrically interpreted as the product of the magnitudes of the two vectors and the cosine of the angle between them. It is defined algebraically as the sum of the products of the corresponding entries of the two sequences of numbers.
In the context of the given exercise, the dot product is used to multiply the vector \( \underline{x} \) with its gradient \( \operatorname{grad} \varphi \), leading to the equation \( \underline{x} \cdot \operatorname{grad} \varphi = m \varphi(\underline{x}) \). The significance of the dot product in this situation is that it collapses the vector information into a scalar, which allows for the comparison between the original function and its rate of change.
Furthermore, dot products are omnipresent in physics and engineering, being used to calculate work done by a force, to define the orthogonality of vectors, and to project one vector onto another.