Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\mathbf{F}=M \mathbf{i}+N \mathbf{j}\) and \(\partial M / \partial x=\partial N / \partial y,\) then \(\mathbf{F}\) is conservative.

Short answer

The given statement is false because, for a vector field to be conservative, it should satisfy \(\partial N/\partial x = \partial M/\partial y\), not \(\partial M/\partial x = \partial N/\partial y\).

Step-by-step solution

Check the statement

Check whether the statement \(\partial M / \partial x = \partial N / \partial y\) is true for the given vector field \(\mathbf{F} = M\mathbf{i}+N\mathbf{j}\). This involves comparing the partial derivatives of the vector field components.

Analyze the condition

Consider the defining property of a conservative field. A vector field \(\mathbf{F} = M \mathbf{i} + N \mathbf{j}\) is considered conservative if and only if \(\partial M/\partial y = \partial N/\partial x\), not \(\partial M/\partial x = \partial N/\partial y\). So, check if the latter condition is actually met.

Conclude the analysis

Since the condition for \(\mathbf{F}\) to be conservative is \(\partial N/\partial x = \partial M/\partial y\), not \(\partial M/\partial x = \partial N/\partial y\), one can conclude that the original statement is false.

Key concepts

Partial Derivatives

When delving into the world of vector fields in multivariable calculus, the concept of partial derivatives emerges as a cornerstone. Partial derivatives are essentially the rates of change of a multivariable function in regards to one variable while keeping the other variables constant. Think of it like analyzing the slope of a terrain in just the east-west direction, ignoring the north-south inclination for a moment.

Mathematically, for a function f(x, y), the partial derivative with respect to x is denoted by \( \frac{\partial f}{\partial x} \) and illustrates how the function f changes as x varies and y is held fixed. Similarly, \( \frac{\partial f}{\partial y} \) represents the change in f as y changes with x remaining constant. In the context of vector fields, when \( \mathbf{F} = M\mathbf{i} + N\mathbf{j} \) where M and N are functions of both x and y, \( \frac{\partial M}{\partial x} \) and \( \frac{\partial N}{\partial y} \) become especially significant, as they play a key role in determining certain properties of the field, such as whether it is conservative or not.

Vector Fields in Calculus

Vector fields allow us to visualize and understand multi-dimensional flows, like wind patterns across a landscape or the electric field around a charged object. They are defined as functions that assign a vector to each point in space. In the context of two-dimensional space, a vector field \( \mathbf{F} \) can be represented as \( \mathbf{F} = M\mathbf{i} + N\mathbf{j} \) where M and N are the components of the field in the x and y directions, respectively.

A particularly interesting type of vector field is the conservative vector field. Conservative fields have the special property that the work done to move a particle from one point to another is path-independent; the same energy is expended no matter what path is taken between the points. This is tied to the concept of a potential function, a scalar function whose gradient gives us back the original vector field. To check if a vector field is conservative, we can employ the test using partial derivatives, which is mentioned in the exercise — conservative fields must satisfy the condition where \( \frac{\partial N}{\partial x} = \frac{\partial M}{\partial y} \) everywhere in the field.

Mathematical Proof

In mathematics, a proof is a logical argument that establishes the truth or falsity of a statement, using a series of deduced conclusions from known truths, axioms, and theorems. To 'prove' something within the realms of mathematics means to convincingly demonstrate that it must be true in all conceivable cases within the defined parameters.

Taking the given exercise as an example, the statement proposed needs to be verified through a mathematical proof. The proof involves verifying the condition for a vector field to be conservative. By examining the partial derivatives \( \frac{\partial M}{\partial x} \) and \( \frac{\partial N}{\partial y} \) and checking if they satisfy the correct equality, one can either affirm or negate the conservativeness of the vector field \( \mathbf{F} \). As highlighted in the solution, the observed vector field in the exercise does not satisfy the necessary condition \( \frac{\partial N}{\partial x} = \frac{\partial M}{\partial y} \) for conservativeness, hence, the statement is disproved. This logical deduction, supported by knowledge of vector calculus, renders the conclusion that the field is not conservative.