Question

If \({\rm{F = Pi + Qj}}\). how do you test to determine whether F is conservative? What if F is a vector field on \({\mathbb{R}^3}\)?

Short answer

If the curl \({\rm{F = 0}}\), then the vector field F is conservative.

Step-by-step solution

Definition of Concept

Velocity: The rate of travel of an object, as well as its direction, is defined as velocity. The velocity of an object tells you how fast it is moving and in which direction it is moving. It's similar to speed, but with the addition of direction. While speed is a directionless quantity, velocity is a directional quantity.

Find thevalue of curl F

If and only if, the vector field\({\rm{F = Pi + Qj}}\)is conservative.

\(\frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}\)

If and only if curl\({\rm{F = 0}}\), the vector field F on\({\mathbb{R}^3}\)is conservative.

By definition curl\({\bf{F}} = \nabla \times {\bf{F}}\)

If\({\rm{F = Pi + Qj + Rk}}\), then if and only if, F is conservative.

\(\left| {\begin{aligned}{{}{}}{\bf{i}}&{\bf{j}}&{\bf{k}}\\{\frac{\partial }{{\partial x}}}&{\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial z}}}\\{{\rm{ text }}}&Q&R\end{aligned}} \right| = 0\)

Which is\(\left( {\frac{{\partial R}}{{\partial y}} - \frac{{\partial Q}}{{\partial z}}} \right){\bf{i}} + \left( {\frac{{\partial P}}{{\partial z}} - \frac{{\partial R}}{{\partial x}}} \right){\bf{j}} + \left( {\frac{{\partial Q}}{{\partial x}} - \frac{{\partial P}}{{\partial y}}} \right){\bf{k}} = 0\)

Which means that F is conservative if and only if the conditions below are met.

\(\begin{aligned}{}\frac{{\partial R}}{{\partial y}} &= \frac{{\partial Q}}{{\partial z}}\quad {\rm{ }}\\{\rm{and }}\quad \frac{{\partial P}}{{\partial z}}& = \frac{{\partial R}}{{\partial x}}\quad {\rm{ }}\\{\rm{and }}\quad \frac{{\partial Q}}{{\partial x}}& = \frac{{\partial P}}{{\partial y}}\end{aligned}\)

Therefore, if the curl \({\rm{F = 0}}\), then the vector field F is conservative.