Question

Find the most general function \(f(\mathbf{p})\) satisfying \(\nabla f(\mathbf{p})=\mathbf{p}\).

Short answer

The most general function is \(f(\mathbf{p}) = \frac{x_1^2}{2} + \frac{x_2^2}{2} + \ldots + \frac{x_n^2}{2} + C\), where \(C\) is any constant.

Step-by-step solution

Understand the Gradient

The equation \(abla f(\mathbf{p}) = \mathbf{p}\) tells us that the gradient of the function \(f(\mathbf{p})\) is equal to the vector \(\mathbf{p}\). This suggests that \(f(\mathbf{p})\) is a scalar field with a gradient exactly equal to \(\mathbf{p}\).

Define the Vector and Its Components

Suppose \(\mathbf{p} = (x_1, x_2, \ldots, x_n)\). This means the gradient \(abla f(\mathbf{p})\) consists of partial derivatives \(\left( \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \ldots, \frac{\partial f}{\partial x_n} \right)\) that equal each corresponding component.

Integrate Each Component Independently

For each component \(x_i\), integrate \(\frac{\partial f}{\partial x_i} = x_i\) with respect to \(x_i\). Each integral yields \(f(x_1, x_2, \ldots, x_n) = \int x_i \, dx_i = \frac{x_i^2}{2} + C(x_1, x_2, ..., \hat{x}_i, ..., x_n)\), where \(\hat{x}_i\) indicates that \(x_i\) is not a variable of the constant \(C\).

Determine the General Solution

Sum the results of each integral to obtain the overall function. This gives \(f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^n \frac{x_i^2}{2} + C\), where \(C\) can be any differentiable function that depends only on combinations of the variables that were not involved in the differentiation, which can be essentially any constant with no dependence on variables or additional function of the remaining variables.

Verify the Solution

Check the solution by computing the gradient \(abla f\) of the function \(f(x_1, x_2, \ldots, x_n) = \sum \frac{x_i^2}{2} + C\). The gradient yields \(\left( x_1, x_2, \ldots, x_n \right)\), which confirms \(abla f = \mathbf{p}\), validating that the found function is correct.

Key concepts

Scalar Field

A scalar field is a mathematical function that assigns a single value or scalar to every point in space. In this context, the function \( f(\mathbf{p}) \) is a scalar field of the vector \( \mathbf{p} \). Each point in space is represented by the vector \( \mathbf{p} = (x_1, x_2, \ldots, x_n) \), and the scalar field assigns a single value to each point. - Scalar fields are often used to describe physical quantities that have magnitude but no direction, like temperature or pressure.- In the exercise, \( f(\mathbf{p}) \) is such a field, and is defined in terms of its gradient, which is given as equal to \( \mathbf{p} \). Understanding scalar fields involves grasping how they change and how this change is represented through gradients, which naturally leads us to partial derivatives.

Partial Derivatives

Partial derivatives are crucial in understanding how functions change with respect to one variable at a time, while holding other variables constant. In this exercise, partial derivatives help us understand the components of the gradient. When we take the gradient \( abla f(\mathbf{p}) \) of the function \( f \), it results in a vector whose components are partial derivatives of \( f \). Specifically, we have:

• \( \frac{\partial f}{\partial x_1} = x_1 \)
• \( \frac{\partial f}{\partial x_2} = x_2 \)
• ... up to \( x_n \)

These derivatives represent how the function \( f \) changes as each component of the vector \( \mathbf{p} \) changes. By calculating these, we effectively examine the influence of each variable independently.

Integration of Vector Components

Integration is the reverse process of differentiation. To determine the actual function \( f(\mathbf{p}) \), we need to integrate each component of the gradient. This involves integrating each partial derivative with respect to its corresponding variable. In mathematical terms, we perform an integral for each \( x_i \):- \( \int x_i \, dx_i = \frac{x_i^2}{2} + C(x_1, x_2, \ldots, \hat{x}_i, \ldots, x_n) \) The expression \( C(x_1, x_2, \ldots, \hat{x}_i, \ldots, x_n) \) represents the constant of integration, which may depend on all other variables except \( x_i \). This integration process gives us potential components of the original scalar field \( f \). By summing these integrated components, we move towards the full general solution where each directional "slice" of the field has been accounted for.

General Solution of Differential Equations

The general solution involves summing the integrated components, revealing a function that satisfies the differential equation \( abla f = \mathbf{p} \). Upon performing integrations, the accumulated function is:\[ f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^n \frac{x_i^2}{2} + C \]Here, \( C \) stands for any differentiable function of the complementary variables not included in the gradient. What makes it the general solution is the flexibility of \( C \), which can take any form as long as it satisfies the conditions of differentiation.If you recompute the gradient of this generalized function, it must equate to \( \mathbf{p} \), confirming the solution's validity. This is crucial in differential equations as it captures all potential forms of solutions enveloped within a single expression, considering every allowable combination of the constants.