Problem 1 - Thomas Calculus

Question

Which fields are conservative, and which are not? \(\mathbf{F}=y z \mathbf{i}+x z \mathbf{j}+x y \mathbf{k}\)

Step-by-Step Solution

Answer

The vector field \( \mathbf{F} = yz \mathbf{i} + xz \mathbf{j} + xy \mathbf{k} \) is conservative.

Step 1: Understand the Definition of a Conservative Vector Field

A vector field \( \mathbf{F} \) is conservative if there exists a scalar potential function \( \phi \) such that \( \mathbf{F} = abla \phi \). This also implies that the curl of the field is zero, i.e., \( abla \times \mathbf{F} = \mathbf{0} \).

Step 2: Compute the Curl of \( \mathbf{F} \)

Given \( \mathbf{F} = yz \mathbf{i} + xz \mathbf{j} + xy \mathbf{k} \), the curl \( abla \times \mathbf{F} \) is calculated as follows using the determinant:\[abla \times \mathbf{F} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \yz & xz & xy\end{vmatrix}\]Compute each component of the curl.

Step 3: Calculate the \( \mathbf{i} \)-component

For the \( \mathbf{i} \)-component, compute:\[\frac{\partial (xy)}{\partial y} - \frac{\partial (xz)}{\partial z} = x - x = 0\]

Step 4: Calculate the \( \mathbf{j} \)-component

For the \( \mathbf{j} \)-component, compute:\[\frac{\partial (yz)}{\partial z} - \frac{\partial (xy)}{\partial x} = y - y = 0\]

Step 5: Calculate the \( \mathbf{k} \)-component

For the \( \mathbf{k} \)-component, compute:\[\frac{\partial (xz)}{\partial x} - \frac{\partial (yz)}{\partial y} = z - z = 0\]

Step 6: Determine if the Vector Field is Conservative

The result of \( abla \times \mathbf{F} \) is \( \mathbf{0} \) for all components, indicating that the vector field \( \mathbf{F} \) is conservative. If any component of the curl was not zero, \( \mathbf{F} \) would not be conservative.

Key Concepts

Vector Calculus

Vector Calculus is a branch of mathematics that focuses on vector fields and operations that can be performed on them. A vector field is a function that assigns a vector to each point in space. In physics and engineering, vector fields are often used to represent phenomena such as force fields or velocity fields, where both magnitude and direction are important.

Key operations in vector calculus include the gradient, divergence, and curl. These operations help describe and analyze the behavior and properties of vector fields. The gradient provides the rate and direction of change in a scalar field, divergence represents the magnitude of a source or sink at a given point, and curl captures the rotation of a vector field.

In this context, understanding these operations can help in determining whether a vector field is conservative. A conservative vector field has certain properties, such as having a zero curl, which are essential in simplifying complex physical and mathematical problems. Techniques from vector calculus are also used in line integrals and surface integrals, further broadening its applications.

Curl of a Vector Field

The curl of a vector field is a crucial concept in vector calculus, used to measure the field's rotation or "curling" tendency at a point. Mathematically, the curl of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) in three-dimensional space is represented as the cross product:

• \( abla \times \mathbf{F} = \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \mathbf{i} + \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \mathbf{j} + \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \mathbf{k} \)

When computing the curl, we often use the determinant of a 3x3 matrix made from unit vectors, partial derivatives, and the components of the vector field.

A vector field is considered curl-free, or irrotational, if the curl is zero everywhere in the field. This property is important because it indicates that the vector field could be expressed as the gradient of some scalar potential function. In practical terms, the absence of curl implies that movement within the field is path-independent, which often simplifies calculations in physics and engineering.

Scalar Potential Function

A scalar potential function is a vital concept when dealing with conservative vector fields. If a vector field \( \mathbf{F} \) is conservative, then there exists a scalar potential \( \phi \) such that \( \mathbf{F} = abla \phi \). In simple terms, this means the vector field can be described as the gradient of the scalar potential function.

One of the significant characteristics of a conservative vector field is path-independence, meaning that the work done by the field along two different paths between the same two points is the same. This property allows the field to be described solely by its potential function \( \phi \), providing a simplified way to analyze it.

• To find the potential function, each component of the vector field is integrated with respect to its respective variable.

Finding a potential function involves integrating the components of the vector field. Once obtained, it provides valuable insights into the physical or geometrical interpretations of the field. In practical terms, knowing the potential function helps in solving problems related to fields like gravitational, electrostatic, or fluid dynamics where energy conservation is key.