Q. 15 - Calculus

Question

How would you show that a given vector field in ${\mathrm{ℝ}}^2$ is not conservative?

Step-by-Step Solution

Answer

A given vector field in ${\mathrm{ℝ}}^2$ is not conservative when,  $\mathrm{F}(x,y)\ne \nabla f(x,y)\ne \frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}\mathbf{j}$

Step 1.Given Information

How would you show that a given vector field in ${\mathrm{ℝ}}^2$ is not conservative?

Step 2. A vector field is conservative

If a vector field can be described as a gradient, it offers a number of mathematically appealing properties, including the ability to be easily integrated along curves. Such fields are referred to as conservative vector fields.

A vector field that is conservative. F is a vector field that represents the gradient of a function $f$.

Step 3. A conservative vector field F is a vector field that can be written as the gradient of some function f .

That is,

$\mathrm{F}(x,y)=\nabla f(x,y)=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}\mathbf{j}$

Hence, we can say that a given vector field in ${\mathrm{ℝ}}^2$ is not conservative, when

$\mathrm{F}(x,y)\ne \nabla f(x,y)\ne \frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}\mathbf{j}$