Question

a) Show that if \(\mathbf{v}\) is one solution of the equation curl \(\mathbf{v}=\mathbf{u}\) for given \(\mathbf{u}\) in a simply connected domain \(D\), then all solutions are given by \(\mathbf{v}+\operatorname{grad} f\), where \(f\) is an arbitrary differentiable scalar in \(D\). b) Find all vectors \(\mathbf{v}\) such that curl \(\mathbf{v}=\mathbf{u}\) if $$ \mathbf{u}=\left(2 x y z^{2}+x y^{3}\right) \mathbf{i}+\left(x^{2} y^{2}-y^{2} z^{2}\right) \mathbf{j}-\left(y^{3} z+2 x^{2} y z\right) \mathbf{k} $$

Short answer

Answer: The general solution for all vector fields $\mathbf{v}$ such that curl $\mathbf{v} = \mathbf{u}$ is given by: $$ \mathbf{v} = (0, y^2 z^2, -y^2 z^2) + \operatorname{grad} f $$ where $f$ is an arbitrary differentiable scalar function in the domain $D$.

Step-by-step solution

Prove the general solution form

Let \(\mathbf{v}\) and \(\mathbf{w}\) be two solutions of the equation curl \(\mathbf{v} = \mathbf{u}\) in the simply connected domain \(D\). Then, we have curl \(\mathbf{v} = \mathbf{u}\) and curl \(\mathbf{w} = \mathbf{u}\). Subtract the two equations to get: $$ \operatorname{curl} (\mathbf{v} - \mathbf{w}) = 0 $$ Now, we know that in a simply connected domain, if the curl of a vector field is zero, then the vector field can be expressed as the gradient of a scalar function. So, we can write: $$ \mathbf{v} - \mathbf{w} = \operatorname{grad} f $$ for some differentiable scalar function \(f\). Then, all the solutions can be written as \(\mathbf{v} + \operatorname{grad} f\), where \(\mathbf{v}\) is a specific solution, and \(f\) is any differentiable scalar function in the domain \(D\). This completes the proof.

Calculate the curl of the given vector field

Now, let's find all vector fields \(\mathbf{v}\) such that curl \(\mathbf{v} = \mathbf{u}\), where $$ \mathbf{u} = \left(2 x y z^{2}+x y^{3}\right) \mathbf{i}+\left(x^{2} y^{2}-y^{2} z^{2}\right) \mathbf{j}-\left(y^{3} z+2 x^{2} y z\right) \mathbf{k}. $$ First, we need to calculate the curl of this vector field. The curl of a vector field \(\mathbf{A} = (A_x, \ A_y, \ A_z)\) is given by: $$ \operatorname{curl} \mathbf{A} = \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z},\ \frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x},\ \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right). $$ Applying this to the given vector field, we have: $$ \operatorname{curl} \mathbf{u} = \left(\frac{\partial (-\left(y^{3} z+2 x^{2} y z\right))}{\partial y} - \frac{\partial \left(x^{2} y^{2}-y^{2} z^{2}\right)}{\partial z}, \frac{\partial \left(2 x y z^{2}+x y^{3}\right)}{\partial z} - \frac{\partial (-\left(y^{3} z+2 x^{2} y z\right))}{\partial x}, \frac{\partial \left(x^{2} y^{2}-y^{2} z^{2}\right)}{\partial x} - \frac{\partial \left(2 x y z^{2}+x y^{3}\right)}{\partial y}\right). $$ Calculating partial derivatives, we get: $$ \operatorname{curl} \mathbf{u} = \left(-2 y z^{2},\ 2 y^{2} z,\ 0\right). $$ So our desired equation is curl \(\mathbf{v}=( -2 y z^{2},\ 2 y^{2} z,\ 0 )\).

Integrate the curl to find a specific solution

Now, we will use the equation \(\operatorname{curl} \mathbf{v}=( -2 y z^{2},\ 2 y^{2} z,\ 0 )\) to find a specific solution \(\mathbf{v} = (v_x, v_y, v_z)\). We know that: $$ \begin{cases} \frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z}= -2 y z^{2}\\ \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x}= 2 y^{2} z\\ \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y}= 0 \end{cases}. $$ Integrating each equation one variable at a time, we have: - \(\int (-2 y z^{2})\, d y = -y^2 z^{2} + c(z)\) - \(\int (2 y^2 z)\, d z = y^2 z^2 + c(x)\) - \(\int (0)\, d x = c(y)\) +\(c_0\). Now, notice that \(\mathbf{v} = (0, y^2 z^2, -y^2 z^2)\) is a solution to the system. To verify, let's compute the curl: $$ \operatorname{curl} (0, y^2 z^2, -y^2 z^2) = ( -2 y z^2,\ 2 y^2 z,\ 0 ). $$ We see that this is the desired curl.

Write the general solution

Finally, using the result from step 1, we can write the general solution as: $$ \mathbf{v} = (0, y^2 z^2, -y^2 z^2) + \operatorname{grad} f $$ where \(f\) is an arbitrary differentiable scalar function in the domain \(D\). This is the general solution for \(\mathbf{v}\) such that curl \(\mathbf{v} = \mathbf{u}\).

Key concepts

Curl Operation

Understanding the curl operation is crucial in vector calculus. Curl provides information about the rotation of a vector field in three-dimensional space.
The curl of a vector field \(\mathbf{A} = (A_x, A_y, A_z)\) is calculated using the cross-product of the del operator (\(abla\)) with the vector field, resulting in a new vector field:

• The \(x\)-component is given by \(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\).
• The \(y\)-component is \(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\).
• The \(z\)-component is \(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\).

The curl essentially measures the tendency of particles in the vector field to spin around a given axis. When you calculate the curl of a field, and you get zero, it indicates no rotational tendency. In our exercise, understanding the curl helps us transform the vector problem into understanding the potential rotationality of the field \(\mathbf{u}\).

Gradient Field

The gradient of a scalar field \(f\) is a fundamental concept in vector calculus. The "grad" operation (denoted by \(\operatorname{grad} f\) or \(abla f\)) results in a vector field that points in the direction of the greatest rate of increase of the scalar field.
It is composed of the partial derivatives of \(f\):

• The \(x\)-component is \(\frac{\partial f}{\partial x}\).
• The \(y\)-component is \(\frac{\partial f}{\partial y}\).
• The \(z\)-component is \(\frac{\partial f}{\partial z}\).

In our exercise, any vector \(\mathbf{v}\) that is a solution to the equation can be perturbed by adding a gradient field, \(\operatorname{grad} f\), without affecting its curl. This shows the interconnectedness of the solutions through scalar adjustments.
Adding \(\operatorname{grad} f\) does not alter the curl of a vector field, as the curl of a gradient field is always zero.

Simply Connected Domain

A simply connected domain is a fascinating concept in topology and vector calculus. It refers to a region in space with no holes through which loops can break free.
Imagine a stretchable loop in the domain; if it can be shrunk to a point without leaving the domain, the domain is called simply connected.
This concept is crucial for several mathematical theorems, including the one that mentions that the curl of a vector field being zero implies the field is a gradient of some scalar function.
When we say that a domain is simply connected, it helps us make some critical conclusions in our problem: specifically, if \(\operatorname{curl} \mathbf{v} = \operatorname{curl} \mathbf{w}\), then \(\mathbf{v} - \mathbf{w}\) must be gradient of some scalar function in such a domain. Understanding this property allows us to find series of solutions for vector field equations.

Vector Fields

Vector fields are mathematical constructions used to associate a vector to every point in space. Each vector in a vector field represents a quantity with both magnitude and direction, making them vital in physics and engineering.
Imagine wind patterns over a geographical area; the direction and strength of wind at any point are represented by vectors.
In our specific problem, the given \(\mathbf{u}\) describes the changes or differential effects at any point in space in terms of three components.

• \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are unit vectors in the \(x\), \(y\), and \(z\) directions, respectively.
• Each component is a function of \(x\), \(y\), and \(z\), representing how these directions contribute to the vector at any point.

Vector fields are closely associated with differential operations (such as curl and gradient) that help us analyze how these fields behave or change across space. By exploring the behavior of \(\mathbf{u}\) through its curl, we can determine the potential underlying dynamics for systems modeled by vector fields.