Problem 121 - Calculus Volume 3

Question

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. $$ \mathbf{F}(x, y, z)=\left(\frac{1}{y}\right) \mathbf{i}+\left(\frac{x}{y^{2}}\right) \mathbf{j}+(2 z-1) \mathbf{k} $$

Step-by-Step Solution

Answer

The vector field is not conservative as its curl is non-zero.

Step 1: Verify if the Vector Field is Conservative

Check if the vector field \( \mathbf{F} = \left( \frac{1}{y} \right) \mathbf{i} + \left( \frac{x}{y^2} \right) \mathbf{j} + (2z - 1) \mathbf{k} \) is conservative. A vector field is conservative if its curl is zero, i.e., \( abla \times \mathbf{F} = \mathbf{0} \).

Step 2: Compute the Curl of the Vector Field

Calculate the curl of \( \mathbf{F} \) using the formula for curl: \( abla \times \mathbf{F} = \left( \frac{\partial F_k}{\partial y} - \frac{\partial F_j}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_i}{\partial z} - \frac{\partial F_k}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_j}{\partial x} - \frac{\partial F_i}{\partial y} \right) \mathbf{k} \).

Step 3: Compute Partial Derivatives for Curl

Find the necessary partial derivatives: \( \frac{\partial F_k}{\partial y} = 0 \) and \( \frac{\partial F_j}{\partial z} = 0 \), so the \( \mathbf{i} \) component of the curl is \( 0 - 0 = 0 \).\( \frac{\partial F_i}{\partial z} = 0 \) and \( \frac{\partial F_k}{\partial x} = 0 \), so the \( \mathbf{j} \) component is \( 0 - 0 = 0 \).\( \frac{\partial F_j}{\partial x} = \frac{1}{y^2} \) and \( \frac{\partial F_i}{\partial y} = -\frac{1}{y^2} \), so the \( \mathbf{k} \) component is \( \frac{1}{y^2} - \left(-\frac{1}{y^2}\right) = \frac{2}{y^2} \).

Step 4: Interpret the Result of the Curl

The curl of \( \mathbf{F} \) is \( (0)\mathbf{i} + (0)\mathbf{j} + \left(\frac{2}{y^2}\right)\mathbf{k} \). Since the curl is not equal to zero, the vector field is not conservative.

Key Concepts

Curl of a Vector Field

In vector calculus, the **curl of a vector field** is a powerful tool used to determine if a vector field is conservative. A vector field is said to be conservative if the curl of the vector field is zero. The curl itself is a vector quantity that measures the tendency to rotate about a point in a vector field. To compute the curl of a vector field \( \mathbf{F}(x, y, z) = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), we use the determinant format: \[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix} \]This involves partial derivatives and applies to a three-dimensional vector field. If the curl does not result in the zero vector, then the field has some rotation and is therefore not conservative.

Potential Function

A **potential function** is a scalar function whose gradient is equal to a given vector field. The existence of a potential function means that the vector field is conservative. This implies that there is no net work done in moving along a closed path.In formulas, if a vector field \( \mathbf{F} \) is conservative, there exists a function \( f \) such that \( abla f = \mathbf{F} \). Consequently, if we can find such an \( f \), we have the potential function.Finding a potential function involves solving the system given by the gradient conditions, thus connecting directly to the concept of partial derivatives.

Partial Derivatives

**Partial derivatives** are essential in the discussion of vector fields and particularly when computing the curl. They represent how a function changes as each individual variable changes while holding the other variables constant.For a function \( z = f(x, y) \), the partial derivatives \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) measure its rate of change in the directions of \( x \) and \( y \), respectively. In vector calculus, these derivatives are used to compute the components of the curl.By finding each component this way, one can accurately assess the vector field's properties and determine behaviors such as rotation and potential conservatism.

Vector Calculus

**Vector calculus** is a branch of mathematics focussed on vector fields and differentiable functions. It involves various operations like differentiation, integration, and finding quantities like the curl. By using vector calculus, you can explore physical phenomena, such as fluid flow or electromagnetic fields, with vectors representing quantities that have both direction and magnitude. Key operations like calculating the gradient, divergence, and curl simplify the process of modeling and solving real-world problems. Understanding these operations allows deeper insights into fields' behaviors, such as identifying whether a particular field is conservative or understanding the implications of its curl.