Question

Sind die Vektorfelder wirbelfrei? a) \(\vec{F}(x, y, z)=(a, x, b)\) b) \(\vec{F}(x, y, z)=\frac{(x, y, z)}{x^{2}+y^{2}+z^{2}}\)

Short answer

Both vector fields, \(\vec{F}(x, y, z)=(a, x, b)\) and \(\vec{F}(x, y, z)=\frac{(x, y, z)}{x^{2}+y^{2}+z^{2}}\), are curl free.

Step-by-step solution

Computation of the Curl - Part a

Let's compute the Curl of \(\vec{F}(x, y, z)=(a, x, b)\). The Curl is calculated as: \(Curl(F)=\(\nabla \times \vec{F} = \left( \frac{ \partial b}{\partial y}-\frac{ \partial x}{\partial z}, \frac{ \partial a}{\partial z}-\frac{ \partial a}{\partial x}, \frac{ \partial x}{\partial y}-\frac{ \partial a}{\partial x} \right)\). Since \(a\) and \(b\) are constant and \(x\), \(y\), \(z\) are independent, the partial derivatives yield zero.

Interpretation of the Result - Part a

Since the Curl of the vector field \(\vec{F}(x, y, z)=(a, x, b)\) equals zero, this vector field is curl-free or irrotational.

Computation of the Curl - Part b

Now, let's compute the Curl of \(\vec{F}(x, y, z)=\frac{(x, y, z)}{x^{2}+y^{2}+z^{2}}\). Again, \(Curl(F)= \nabla \times \vec{F}\), but in this case, the determination involves more complex derivatives. After some computation, we find that \(Curl(F)=\left(0,0,0\right)\).

Interpretation of the Result - Part b

Since the Curl of the vector field \(\vec{F}(x, y, z)=\frac{(x, y, z)}{x^{2}+y^{2}+z^{2}}\) equals zero as well, this vector field is also curl-free or irrotational.

Key concepts

Understanding Irrotational Vector Fields

A vector field is termed "irrotational" if its curl is zero throughout the entire field. In simpler terms, an irrotational vector field does not have any local rotations; that is, the small circulation around any point in the field is zero.
This property is significant in ensuring that the vector field behaves predictably in many physical and mathematical scenarios where rotation is unwanted or needs to be minimized.

• When the curl of a vector field is zero, it suggests no "swirling" or "vortex-like" behavior within the field.
• Such fields can often be represented as the gradient of some scalar potential function, making them conservative.

Understanding irrotational fields helps in fluid dynamics and electromagnetism, where such fields often describe the absence of rotation within a flow or electromagnetic field.

Exploring Partial Derivatives in Curl Computation

Partial derivatives are fundamental in computing the curl of a vector field. They represent how the field changes in response to small variations in each direction—x, y, and z.
This is crucial when deriving exact values for each component of the curl, as follows in the standard mathematical vector calculus formula for curl:

• To compute the curl, you'll look at mixed partial derivatives like \( \frac{ \partial F_3}{\partial y} - \frac{ \partial F_2}{\partial z} \), which elucidate how changes in one direction relate to another.
• This involves working with each component of the given vector field independently to capture the intricate ways those components shift.

Mastering partial derivatives is not only key to calculating curls but is also a stepping stone to deeper vector calculus concepts.

Key Ideas in Vector Calculus

Vector calculus is an essential field of mathematics that focuses on the application of calculus to vector fields. Through this, we aim to understand how to manipulate fields in multi-dimensional spaces.
It combines techniques and concepts from calculus and linear algebra to provide tools to analyze physical phenomena.

• This includes operations such as divergence, gradient, and curl, which help in capturing diverse attributes of vector fields.
• In practical terms, these operations help determine how vector fields behave at a regional or point level, within any given space.

Grasping these notions is critical for advancing in fields like fluid mechanics or electromagnetism, where vector fields commonly describe the distribution of velocities or electromagnetic forces.

Applying Mathematical Physics in Understanding Vector Fields

Mathematical physics harnesses mathematics to explore and explain physical phenomena. In the case of vector fields, it enables a structured approach to understanding spatial distributions and movements of physical quantities such as electromagnetic fields or fluid flow.
With the application of vector calculus in mathematical physics, problems that describe physical systems can be precisely formulated and solved using mathematical languages and methods.

• Vector fields describe quantities that have both direction and magnitude—such as wind velocity or magnetic fields—through mathematical vectors.
• The concept of curl is often key in evaluating whether a field might be experiencing local rotation or circulation, which is vital in physics for describing more dynamic situations.

Using mathematical physics principles strengthens our understanding of the physical world by translating observations into mathematical problems and solutions.