Question

Let \(\mathbf{u}=\frac{-y}{x^{2}+y^{2}} \mathbf{i}+\frac{x}{x^{2}+y^{2}} \mathbf{j}+z \mathbf{k}\) and let \(D\) be the interior of the torus obtained by rotating the circle \((x-2)^{2}+z^{2}=1, y=0\) about the \(z\)-axis. Show that curl \(\mathbf{u}=0\) in \(D\) but

Short answer

#tag_title# Step 2: Expand the determinant and compute the partial derivatives. #tag_content# To find the curl $\mathbf{u}$, let's expand the determinant and compute the partial derivatives: Curl $\mathbf{u} = \left(\frac{\partial}{\partial y}(z) - \frac{\partial}{\partial z}\left(\frac{x}{x^{2}+y^{2}}\right)\right)\mathbf{i} - \left(\frac{\partial}{\partial x}(z) - \frac{\partial}{\partial z}\left(-\frac{y}{x^{2}+y^{2}}\right)\right)\mathbf{j} + \left(\frac{\partial}{\partial x}\left(\frac{x}{x^{2}+y^{2}}\right) - \frac{\partial}{\partial y}\left(-\frac{y}{x^{2}+y^{2}}\right)\right)\mathbf{k}$ Evaluating the partial derivatives, we have: Curl $\mathbf{u} = (0 - 0)\mathbf{i} - (0 - 0)\mathbf{j} + \left(\frac{y^2 - x^2}{(x^2 + y^2)^2} + \frac{x^2 - y^2}{(x^2 + y^2)^2}\right)\mathbf{k}$ Simplifying the last term: Curl $\mathbf{u} = \left(\frac{y^2 - x^2 + x^2 - y^2}{(x^2 + y^2)^2}\right)\mathbf{k} = 0\mathbf{k}$ Curl $\mathbf{u} = 0$. #tag_title# Step 3: Check whether the curl is zero in the given domain \(D\). #tag_content# The domain \(D\) is given by $\{(x, y, z) \in \mathbb{R}^3 \ | \ x^2 + y^2 > 0 \}$, which means that the point \((x, y, z)\) is not at the origin (since the inequality is strict). We have found that the Curl $\mathbf{u} = 0$, which is valid for all $(x, y, z) \neq (0, 0, 0)$. Since the domain \(D\) does not include the origin, the curl of the vector field is zero within the domain \(D\). #Conclusion# We have shown that the curl of the given vector field \(\mathbf{u}\) is equal to zero within the given domain \(D\).

Step-by-step solution

Calculate the curl of the vector field \(\mathbf{u}\).

To find the curl of \(\mathbf{u}\), we will perform the following calculation: Curl \(\mathbf{u} = \nabla \times \mathbf{u}\), where \(\nabla = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})\) is the gradient operator. Using the given vector field, \(\mathbf{u}=\frac{-y}{x^{2}+y^{2}} \mathbf{i}+\frac{x}{x^{2}+y^{2}}\mathbf{j}+z \mathbf{k}\), we have: Curl $\mathbf{u} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ -\frac{y}{x^{2}+y^{2}} & \frac{x}{x^{2}+y^{2}} & z \end{vmatrix} $

Key concepts

Vector Calculus

Vector calculus is a branch of mathematics that focuses on differential and integral calculus with vectors in 2 or more dimensions. It's essential for understanding fields such as physics and engineering, where quantities have both magnitude and direction. In vector calculus, you'll encounter operations like differentiation and integration, but applied to vector fields rather than just single-variable functions.

In the context of our exercise, understanding vector fields and their behaviors is crucial. A vector field on a plane, for example, assigns a vector to each point in the plane, illustrating how a particle would move at each location. This concept is deeply tied to the study of fluid flow, electromagnetic fields, and other complex systems.

Curl Operation

The curl operation is a measure of the rotation or 'twisting' of a 3-dimensional vector field at a point. It's portrayed as another vector field showing where and how fast the 'twisting' occurs. Imagine a paddle wheel placed in a fluid flow; the way the wheel spins gives you the curl at that point. In the language of mathematics, the curl of a vector field \(\mathbf{v}\) is denoted as \(abla \times \mathbf{v}\), where \(abla\) is the gradient operator.

The calculation involves a determinant and partial derivatives, as you would have seen in the step-by-step solution. When the result of a curl operation is zero, it indicates that the vector field is irrotational at that point.

Torus

A torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. It resembles a doughnut or a lifebuoy. In the given exercise, the torus is created by rotating a circle around the z-axis.

The mathematical significance of a torus lies in its topology and complex structure. Understanding the properties of a torus is important when you're working with fields encompassed within it, especially when considering the applications in physics and engineering, such as designing magnetic containment fields in fusion reactors.

Gradient Operator

The gradient operator, denoted as \(abla\), is a vector operator that points in the direction of the greatest rate of increase of a scalar field and its magnitude is the rate of increase in that direction. It combines partial differentiation with respect to each coordinate direction.

In the case of the curl operation, we use this operator in a cross product with the vector field, thus yielding the rotation or 'curl' of the vector field. Understanding how to apply this operator is key to solving many problems in vector calculus and in interpreting physical phenomena related to fields.

Advanced Calculus

Advanced calculus, also known as multivariable calculus, extends the concepts of differentiation and integration to functions of several variables. It covers topics such as vector fields, partial derivatives, multiple integrals, line integrals, surface integrals, and the theorems of Green, Stokes, and Gauss. These tools become indispensable when modeling and solving problems in higher dimensions.

In our exercise, advanced calculus concepts such as the curl of a vector field play a crucial role in understanding the behavior of the vector field within the interior of a torus. Mastery of these concepts is essential for fields that require complex mathematical modeling and analysis.