Problem 34 - Calculus: Early Transcendental Functions

Question

Determine whether or not the vector field is conservative. If it is, find a potential function. $$\left\langle z^{2}+2 x y, x^{2}+1,2 x z-3\right\rangle$$

Step-by-Step Solution

Answer

The vector field \( \langle z^{2}+2 x y, x^{2}+1,2 x z-3 \rangle \) is conservative, and its potential function is \( f(x, y, z) = x^{2}y + z^{2}x + y - 3z \).

Step 1: Identifying the Vector Field

First we write down the given vector field, \( F(x, y, z) = \langle z^{2}+2 x y, x^{2}+1, 2 x z-3 \rangle \) in component form, with \( F_{1} = z^{2} + 2xy \), \( F_{2} = x^{2} + 1 \), and \( F_{3} = 2 x z - 3 \).

Step 2: Checking if the Vector Field is Conservative

A vector field is conservative if its curl is zero. We calculate the curl as\[\text{curl}(F) = (\frac{\partial F_{3}}{\partial y} - \frac{\partial F_{2}}{\partial z}, \frac{\partial F_{1}}{\partial z} - \frac{\partial F_{3}}{\partial x}, \frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y}) = (0, 0, 0).\]We observe that curl(F) is zero, hence the vector field is conservative.

Step 3: Finding the Conservative Potential Function

For a conservative vector field, the potential function (\( f \)) exists and can be found by integrating each component of the vector field. We calculate the integral of each function \( F_{1}, F_{2}, F_{3} \) with respect to \( x, y, z \), respectively, and sum them up, skipping constants for now as they can be arbitrary.This gives us:\[\int F_{1} \, dx = x^{2}y + z^{2}x, \, \int F_{2} \, dy = y + x^{2}y, \, \int F_{3} \, dz = xz^{2} -3z.\]Adding them together and removing duplicate terms, we have \( f(x, y, z) = x^{2}y + z^{2}x + y - 3z \).This is the potential function for the given vector field.

Key Concepts

Vector Calculus

Vector calculus is a sophisticated branch of mathematics that deals with vector fields. A vector field assigns a vector to every point in space, often represented as \( F(x, y, z) = \langle F_1(x, y, z), F_2(x, y, z), F_3(x, y, z) \rangle \). Such fields are crucial in physics and engineering for modeling different forces like gravitational, electromagnetic, and fluid dynamics.

Moreover, vector calculus forms the backbone of many mathematical models used in simulations. It allows for the calculation of gradients, divergences, and curls, which provide important information about the behavior of fields in space. Understanding these properties helps to predict how a particular field might behave at different points or regions.

• **Gradient** measures the rate and direction of change in a field.
• **Divergence** quantifies the extent to which a field spreads out from a point.
• **Curl** describes the rotation of a field around a point.
  

Mastering vector calculus is key to unlocking many physical phenomena and solving complex real-world problems effectively.

Potential Functions

When discussing conservative vector fields, potential functions become a central concept. A vector field \( F \) is conservative if it can be expressed as the gradient of a scalar potential function \( f \). This means that \( F = abla f \), where \( abla \) is the gradient operator.

Potential functions are foundational in physics, particularly in fields like electrostatics and gravitational theory. For example, the potential energy in a gravitational field is derived from a potential function.

• **Finding a Potential Function:**
  • Integrate each component of the vector field with respect to its respective variable.
  • Sum the integrals, ensuring that you manage overlapping terms correctly.
  • Often, you can omit constants when integrating since they do not affect the potential nature of the function.
    

The beauty of potential functions is their simplicity in reducing the complexity of vector fields, making analysis and application more straightforward.

Curl of a Vector Field

The curl is a key operator in vector calculus that reveals important information about a field’s rotational behavior. For a vector field \( \mathbf{F} = \langle F_1, F_2, F_3 \rangle \), the curl is computed as:\[\text{curl}(\mathbf{F}) = abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right)\]

This operation is fundamental in determining whether a vector field is conservative. A conservative field has zero curl, meaning it has no rotational component and is path-independent for line integrals. This facet is significant in ensuring physical systems obey conservation of energy.

• **Properties of Curl:**
  • If \( \text{curl}(\mathbf{F}) = \mathbf{0} \), then the vector field is typically conservative.
  • The curl of a gradient field is always zero, reflecting the absence of a rotational component.
    

Understanding curl helps in analyzing fluid flow, electromagnetic fields, and more, making it an indispensable tool in scientific and engineering disciplines.