Question

Linear and quadratic vector fields a. For what values of \(a, b, c,\) and \(d\) is the field \(\mathbf{F}=\langle a x+b y, c x+d y\rangle\) conservative? b. For what values of \(a, b,\) and \(c\) is the field \(\mathbf{F}=\left\langle a x^{2}-b y^{2}, c x y\right\rangle\) conservative?

Short answer

Question: Determine the values of the constants for which the given vector fields are conservative. a. \(F = \langle ax + by, cx + dy \rangle\) b. \(F = \langle ax^2 - by^2, cxy \rangle\) Answer: a. For the linear vector field to be conservative, \(c = b\). b. For the quadratic vector field to be conservative, \(c = -2b\).

Step-by-step solution

Compute the curl of the linear vector field

Recall that the curl of a vector field \(\mathbf{F}=\langle P, Q \rangle\) is defined as \(\nabla \times \mathbf{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\). In this case, \(P = ax + by\) and \(Q = cx + dy\). So, calculating the partial derivatives: $$ \frac{\partial Q}{\partial x} = c, \quad\text{and}\quad \frac{\partial P}{\partial y} = b $$

Check if the curl is zero

Now we have the curl of the linear vector field \(\mathbf{F}\): \(\nabla \times \mathbf{F} = c - b\). In order for the vector field to be conservative, the curl must be zero: $$ c - b = 0 $$ Thus, for the linear vector field \(\mathbf{F}=\langle a x + b y, c x + d y\rangle\) to be conservative, \(c = b\). For part b:

Compute the curl of the quadratic vector field

As before, the curl of a vector field \(\mathbf{F}=\langle P, Q \rangle\) is \(\nabla \times \mathbf{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\). In this case, \(P = ax^2 - by^2\) and \(Q = cxy\). So, calculating the partial derivatives: $$ \frac{\partial Q}{\partial x} = cy, \quad\text{and}\quad \frac{\partial P}{\partial y} = -2by $$

Check if the curl is zero

Now we have the curl of the quadratic vector field \(\mathbf{F}\): \(\nabla \times \mathbf{F} = cy + 2by\). In order for the vector field to be conservative, the curl must be zero: $$ cy + 2by = 0 $$ For this to be true, either \(y=0\), which doesn't give us any information about the values of \(a, b, c\), or \(c + 2b = 0\). Thus, for the quadratic vector field \(\mathbf{F}=\langle ax^2 - by^2, cxy\rangle\) to be conservative, we need \(c = -2b\).

Key concepts

Curl of a Vector Field

Understanding the curl of a vector field is a fundamental concept in vector calculus that is essential when dealing with fluid flow, magnetic fields, and more. Imagine you have a fluid flowing through space, and you want to know if there is any sort of spinning or rotation happening at a point. Mathematically, this 'spin' is what we refer to as the curl.

For a two-dimensional vector field represented as \( \mathbf{F} = \langle P, Q \rangle \), where \( P \) and \( Q \) are the components of \( \mathbf{F} \) in the x and y directions, respectively, the curl is defined as \( abla \times \mathbf{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \).

The curl measures the tendency of particles to rotate around a point in a subregion of the vector field. If the curl is zero, there is no rotation at that point, and the vector field is said to be irrotational or conservative. In physical terms, this means that the work done by the field along a closed path is zero, which is a property that can simplify many physics problems.

Partial Derivatives

In order to compute the curl of a vector field, you need to understand partial derivatives. The partial derivatives of a function of multiple variables are its derivatives with respect to each variable, holding the other variables constant.

For example, consider a function \( f(x, y) \). The partial derivative of \( f \) with respect to \( x \) is written as \( \frac{\partial f}{\partial x} \) and represents the rate at which \( f \) changes as \( x \) increases, while keeping \( y \) constant. Similarly, \( \frac{\partial f}{\partial y} \) represents the rate of change of \( f \) as \( y \) increases, with \( x \) held constant.

These concepts are crucial when working with vector fields, as you often need to analyze how each component of the field varies with respect to all the other dimensions. Partial derivatives give you this exact information, allowing you to conclude whether a field is conservative.

Linear and Quadratic Vector Fields

In mathematics and physics, vector fields are often categorized by the type of functions that define their components. Linear vector fields are those with components that are first-degree polynomials, and for example, the field \( \mathbf{F} = \langle ax + by, cx + dy \rangle \) is a linear vector field because each component is a linear function of \( x \) and \( y \).

A field is conservative if it meets certain conditions, one of which is having a zero curl. From our previous discussion, for the linear field to be conservative, the condition \( c = b \) must be met based on the expression for the curl.On the other hand, quadratic vector fields have components that are second-degree polynomials, like \( \mathbf{F} = \langle ax^2 - by^2, cxy \rangle \). The criteria for a quadratic field to be conservative involves a bit more nuance, as the relation \( c = -2b \) must hold to ensure the curl of the field is zero.

Recognizing whether a vector field is linear or quadratic helps us apply specific methods to determine their properties, like their conservative nature, which simplifies many types of computations in the realm of vector calculus.