Question

Berechne das Potential von \(\underline{V}(\underline{x})=\underline{x} /|\underline{x}|^{2}, \underline{x}=\left[\begin{array}{l}x \\\ y\end{array}\right] \in \mathbb{R}^{2} \backslash\\{0\\}\), falls es existiert.

Short answer

The precise answer will depend on the evaluation of the curl of the vector field and the following integral to find the potential. If the curl calculation does not yield zero, the vector field \( V(\underline{x}) \) does not have a potential.

Step-by-step solution

Checking if the vector field is conservative

Firstly, calculate the curl of the vector field \( \nabla \times \underline{V}(\underline{x}) \). If the result equals zero, the vector field is indeed conservative.

Computing the potential

Integrate the vector field \( V(\underline{x}) \) with respect to \( x \) and \( y \) to compute its potential \( \phi(x, y) \). The potential of a conservative vector field \( V = \nabla \phi \) is given by the line integral \( \phi(x, y) = \int V \cdot d\underline{x} \).

Conclusion

If the vector field is conservative, write out the computed potential \( \phi(x, y) \). If it is not conservative, state that it does not have a potential.

Key concepts

Vector Analysis

Vector analysis is a branch of mathematics that deals with quantities having both magnitude and direction, which are known as vectors. It is a crucial tool in the exploration of various fields such as physics, engineering, and computer science. At the heart of vector analysis is the concept of vector fields, which assigns a vector to every point in space.

Vectors in a field can represent various phenomena: for example, velocity in fluid dynamics or force in electromagnetics. Understanding the behavior of such fields involves several operations, including differentiation and integration but focused on vectors. Among these operations are the gradient, divergence, and curl, which help us to analyze the properties of vector fields.

Conservative Vector Fields

In vector analysis, a conservative vector field is one where the work done in moving a particle from one point to another is independent of the path taken. This property implies that the field has a scalar potential, a function from which the vector field can be derived.

A key feature of conservative vector fields is their path-independence, which makes them particularly handy in solving physics problems involving forces and energy. To determine if a vector field is conservative, you can look for a scalar potential or check whether the curl of the vector field is zero - if it is, you can be sure the field is conservative.

Curl of a Vector Field

The curl of a vector field is a measure of the rotation at a point in the field. Imagine placing a tiny paddle wheel at any point; the curl gives you the tendency of the field to rotate this wheel. Mathematically, it is a vector operation denoted by the symbol \( abla \times \), which represents a cross product of the del operator with the vector field.

If the curl of a vector field is zero everywhere, it indicates that the field is irrotational, which is a necessary (but not sufficient) condition for the field to be conservative. Thus, checking the curl is an essential step in finding the potential function for a vector field.

Line Integrals

A line integral is an integral where the function to be integrated is evaluated along a curve. It plays a significant role in vector analysis, particularly for calculating work done by a force field on a moving object.

In the case of conservative fields, the line integral is directly related to the scalar potential of the field. If you want to compute the potential of a vector field, you would use the line integral of the field along a path, starting from some reference point to your point of interest. The fundamental theorem for line integrals states that if the vector field is conservative, this integral will only depend on the end points of the curve, reaffirming the independence of path characteristic of conservative fields.