Question

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 a, b, c, and d for which the following vector fields are conservative. a) \(\mathbf{F}=\langle a x + b y, c x + d y\rangle\) b) \(\mathbf{F}=\left\langle a x^{2} - b y^{2}, c x y\right\rangle\) Answer: a) For the vector field \(\mathbf{F}=\langle a x + b y, c x + d y\rangle\) to be conservative, the condition is \(c=d\). b) For the vector field \(\mathbf{F}=\left\langle a x^{2} - b y^{2}, c x y\right\rangle\) to be conservative, the condition is \(c = -2b\).

Step-by-step solution

Calculate the curl of the first vector field

Calculate the curl of the given vector field \(\mathbf{F}=\langle a x + b y, c x + d y\rangle\). Using the formula for the curl: $$\begin{aligned} \nabla \times \mathbf{F} &= \begin{vmatrix} \hat{i} &\hat{j} & \hat{k} \\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ a x + b y & c x + d y & 0 \end{vmatrix} \\ &=\left(\frac{\partial 0}{\partial y}-\frac{\partial (c x + d y)}{\partial z}\right)\hat{i}-\left(\frac{\partial 0}{\partial x}-\frac{\partial (a x + b y)}{\partial z}\right)\hat{j}+\left(\frac{\partial (c x + d y)}{\partial x}-\frac{\partial (a x + b y)}{\partial y}\right)\hat{k} \end{aligned}$$

Simplify the curl

Simplify the curl result: $$\begin{aligned} \nabla \times \mathbf{F} &= (0-0)\hat{i}-(0-0)\hat{j}+(\frac{\partial (c x + d y)}{\partial x}-\frac{\partial (a x + b y)}{\partial y})\hat{k} \\ &= \left[(c-d)\hat{k}\right] \end{aligned}$$

Determine the conservative conditions

Set the curl equal to zero and solve for the conditions that make the field conservative: $$\nabla \times \mathbf{F} = \mathbf{0}$$ $$(c-d)\hat{k} = 0\hat{k}$$ Under the condition \(c=d\), the field \(\mathbf{F}=\langle a x + b y, c x + d y\rangle\) is conservative. #Part b#

Calculate the curl of the second vector field

Calculate the curl of the given vector field \(\mathbf{F}=\left\langle a x^{2} - b y^{2}, c x y\right\rangle\). Using the formula for the curl: $$\begin{aligned} \nabla \times \mathbf{F} &= \begin{vmatrix} \hat{i} &\hat{j} & \hat{k} \\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ a x^2 - b y^2 & c x y & 0 \end{vmatrix} \\ &=\left(\frac{\partial 0}{\partial y}-\frac{\partial (c x y)}{\partial z}\right)\hat{i}-\left(\frac{\partial 0}{\partial x}-\frac{\partial (a x^2 - b y^2)}{\partial z}\right)\hat{j}+\left(\frac{\partial (c x y)}{\partial x}-\frac{\partial (a x^2 - b y^2)}{\partial y}\right)\hat{k} \end{aligned}$$

Simplify the curl

Simplify the curl result: $$\begin{aligned} \nabla \times \mathbf{F} &= (0-0)\hat{i}-(0-0)\hat{j}+(\frac{\partial (c x y)}{\partial x}-\frac{\partial (a x^2 - b y^2)}{\partial y})\hat{k} \\ &= \left[(c y + 2 b y)\hat{k}\right] \end{aligned}$$

Determine the conservative conditions

Set the curl equal to zero and solve for the conditions that make the field conservative: $$\nabla \times \mathbf{F} = \mathbf{0}$$ $$(c y + 2 b y)\hat{k} = 0\hat{k}$$ The field \(\mathbf{F}=\left\langle a x^{2} - b y^{2}, c x y\right\rangle\) is conservative under the condition \(c y + 2 b y = 0\). Since we need to find the values of a, b, and c, in this case \(c = -2b\).

Key concepts

Curl of a Vector Field

Understanding the curl of a vector field is fundamental in vector calculus, and it's particularly crucial when identifying whether a field is conservative or not.

The curl measures the rotation of a vector field around a point. Mathematically, it is represented as a new vector field. For a three-dimensional vector field represented by \( \mathbf{F} = \langle P(x, y, z), Q(x, y, z), R(x, y, z) \rangle \), the curl is calculated using the cross-product of the del operator \( abla \) and the vector field \( \mathbf{F} \):
$$abla \times \mathbf{F} = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \frac{\partial}{\partial x}& \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\P & Q & R\end{vmatrix}$$
The result is another vector field whose components are partial derivatives of the functions P, Q, and R.

Interpreting Curl in Conservative Fields

A conservative vector field is one where the integral around any closed path is zero, which implies that no energy is lost when moving along the path. In physical terms, this means there is no net 'circulation' or rotation in the field. Therefore, if the curl of a vector field is zero, that field is likely to be conservative. The exercise given demonstrates this principle by setting the curl equal to zero and solving for the necessary conditions to satisfy this property.

Partial Derivatives

The concept of partial derivatives plays a central role in the calculation of the curl, as well as in many other areas of vector calculus. A partial derivative represents how a function changes as one variable changes, with all other variables held constant. For a function \( f(x, y) \), the partial derivative with respect to \( x \) is denoted by \( \frac{\partial f}{\partial x} \) and likewise for \( y \).

In the context of the exercise, partial derivatives are used to calculate the components of the curl vector. For example, the component in the \( \hat{k} \) direction is the difference between the partial derivative of Q with respect to \( x \) and the partial derivative of P with respect to \( y \). This can be mathematically represented as \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \).

Understanding Partial Derivatives in Conservative Fields

When checking if a vector field is conservative, you compare partial derivatives accordingly. If the mixed second-order partial derivatives are continuous and equal, by Clairaut's theorem, it suggests that the curl should be zero, which is a sign of a conservative field. This is why simplifying the expression for the curl is important – it helps reveal the relationships between the partial derivatives in the vector field.

Vector Calculus

Within the realm of vector calculus, we explore various operations on vector fields, such as gradient, divergence, and curl. These concepts enable us to analyze physical systems in terms of fields rather than just at specific points. Vector fields represent the distribution of vectors across a region of space, which can describe phenomena like fluid flows or electromagnetic fields.

As we've seen with the curl, vector calculus operations can reveal the underlying properties of these fields. When a field is conservative, a property easily tested with the curl, this implies that the field can be written as the gradient of a potential function. This potential function, often denoted \( \phi(x, y, z) \), serves as a scalar representation of the vector field. The conservative nature implies that work done against the field is path-independent and can be fully recovered, which is a cornerstone concept in physics, particularly in energy conservation.

Real-World Applications of Vector Calculus

Vector calculus isn't just theoretical; it has practical applications in engineering, physics, and computer graphics. Understanding how to determine if a field is conservative has implications for solving problems related to force fields in mechanics, electric and magnetic fields in electromagnetism, and even in optimizing paths for robotics and autonomous vehicles.