Problem 107 - Calculus Volume 3

Question

For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function. $$ \mathbf{F}(x, y)=\left(-y+e^{x} \sin y\right) \mathbf{i}+\left[(x+2) e^{x} \cos y\right] \mathbf{j} $$

Step-by-Step Solution

Answer

The vector field is not conservative.

Step 1: Check the condition for conservative fields

\(\mathbf{F}(x,y) = (-y+e^x\sin y)\mathbf{i} + (\ldots)\mathbf{j}\). Check if \(\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}\).

Step 2: Determine result

Compute the partial derivatives and compare. If they are not equal, the field is not conservative.

Key Concepts

Potential Function

A potential function is a scalar function whose gradient results in a given vector field. When dealing with vector fields, such as wind or magnetic fields, the potential function provides a way of understanding the field's origin. If a vector field is conservative, it means it can be represented as the gradient of a potential function. To find a potential function, one looks for a scalar function \( f(x, y) \) such that \( abla f = \mathbf{F} \), where \( abla f \) denotes the gradient of \( f \). It involves integrating the components of the vector field and ensuring these integrations are consistent across dimensional variables. Potential functions are significant in physics, often linking fields and forces with energy.

Gradient

The gradient is a vector operator that denotes how a scalar field changes in space. It represents the rate and direction of change of a function. For a function \( f(x,y) \), the gradient is expressed as: \[ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \]The gradient points in the direction where the function increases most rapidly, and its magnitude tells us how steep the increase is. In physical terms, this can be imagined as the slope of a hill that a ball would roll down. In vector calculus, the gradient is used to find whether a vector field is conservative by showing if it equates to the vector field when diverged from a potential function.

Partial Derivatives

Partial derivatives represent a way to understand how a multi-variable function changes as one of the variables changes, while other variables are held constant. They are fundamental in calculus, especially for functions of several variables. For a function \( f(x, y, z) \), partial derivatives look like this:

• \( \frac{\partial f}{\partial x} \) - change of the function with respect to \( x \)
• \( \frac{\partial f}{\partial y} \) - change with respect to \( y \)
• \( \frac{\partial f}{\partial z} \) - change with respect to \( z \)

In vector fields, partial derivatives are used to check if fields are conservative. If the mixed partial derivatives meet specific criteria, the potential function might exist, indicating a conservative field. This criterion ensures consistent integration of multi-variable functions.

Vector Field

A vector field in mathematics assigns a vector to every point in a space. Imagine you have a grid, and every point on that grid has an arrow attached to it—this is your vector field. The direction and magnitude of the arrow represent different forces, like wind speed and direction. Mathematically, a vector field in two dimensions can be written as \( \mathbf{F}(x, y) = P(x, y)\mathbf{i} + Q(x, y)\mathbf{j} \). Vector fields are used widely in physics to describe force fields such as gravitational, electric, and magnetic fields. Not all vector fields can be described by a potential function. If they can, they're known as conservative fields. Determining whether a vector field is conservative involves checking that certain conditions (like the equality of mixed partial derivatives) are met, a process that hones in on the structure and connectivity of the field itself.