4.3 Conservative vector fields and potential functions

Conservative vector fields and potential functions are key concepts in vector calculus. They simplify complex calculations by allowing us to use scalar functions instead of vector fields. This topic builds on earlier ideas of line integrals and gradient fields.

Understanding these concepts is crucial for physics applications. Conservative fields, like gravity, have path-independent work. Potential functions help us calculate work and energy in these fields more easily, connecting math to real-world phenomena.

Conservative Vector Fields and Potential Functions

Properties and Definitions

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• Conservative vector field $\mathbf{F}$ has the property that the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ is independent of the path $C$ between any two points
  • Only depends on the initial and final points of the path
  • Example: Gravitational field $\mathbf{F}(x, y, z) = -\frac{GMm}{r^2}\hat{r}$
• Potential function $\phi$ is a scalar function whose gradient is equal to the conservative vector field $\mathbf{F} = \nabla \phi$
  • Exists for every conservative vector field
  • Can be used to calculate the work done by $\mathbf{F}$ along any path
• Gradient $\nabla \phi$ of a scalar function $\phi(x, y, z)$ is a vector field whose components are the partial derivatives of $\phi$
  • $\nabla \phi = \left(\frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z}\right)$
  • Represents the direction and rate of steepest ascent of $\phi$

Exact Differentials and Scalar Potentials

• Exact differential $d\phi$ is the differential of a scalar function $\phi(x, y, z)$
  • Can be written as $d\phi = \frac{\partial \phi}{\partial x}dx + \frac{\partial \phi}{\partial y}dy + \frac{\partial \phi}{\partial z}dz$
  • Exists if and only if the vector field $\mathbf{F}$ is conservative
• Scalar potential $\phi$ is a scalar function whose differential is equal to the dot product of the conservative vector field $\mathbf{F}$ and the displacement vector $d\mathbf{r}$
  • $d\phi = \mathbf{F} \cdot d\mathbf{r}$
  • Can be found by integrating the components of $\mathbf{F}$ with respect to the corresponding variables (x, y, z)

Path Independence and the Fundamental Theorem

Path Independence

• Path independence means that the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ is independent of the path $C$ between any two points
  • Holds for conservative vector fields
  • Allows the calculation of work done by $\mathbf{F}$ using only the potential function $\phi$
• Conservative forces, such as gravitational and electrostatic forces, have the property of path independence
  • Work done by a conservative force depends only on the initial and final positions
  • Example: Work done by gravity in moving an object from one height to another is independent of the path taken

Fundamental Theorem of Line Integrals

• Fundamental theorem of line integrals states that for a conservative vector field $\mathbf{F} = \nabla \phi$, the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ is equal to the difference in the potential function $\phi$ between the endpoints of the path $C$
  • $\int_C \mathbf{F} \cdot d\mathbf{r} = \phi(\mathbf{r}_1) - \phi(\mathbf{r}_0)$, where $\mathbf{r}_0$ and $\mathbf{r}_1$ are the initial and final points of the path $C$
  • Allows the calculation of work done by $\mathbf{F}$ using only the values of $\phi$ at the endpoints
• Work-energy principle states that the work done by a conservative force is equal to the negative change in potential energy
  • $W = -\Delta U$, where $W$ is the work done and $\Delta U$ is the change in potential energy
  • Follows from the fundamental theorem of line integrals

Testing for Conservative Vector Fields

Curl Test

• Curl test is a method for determining whether a vector field $\mathbf{F}(x, y, z) = (P, Q, R)$ is conservative
  • Computes the curl of $\mathbf{F}$, defined as $\nabla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)$
  • If the curl is zero everywhere, i.e., $\nabla \times \mathbf{F} = \mathbf{0}$, then $\mathbf{F}$ is conservative
• Necessary condition for a vector field to be conservative is that its curl must be zero
  • If the curl is non-zero at any point, the vector field is not conservative
  • Example: Magnetic field $\mathbf{B}$ is not conservative, as its curl is non-zero ($\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$, where $\mathbf{J}$ is the current density)
• Sufficient condition for a vector field defined on a simply connected domain to be conservative is that its curl is zero everywhere
  • Simply connected domain is a region where any closed loop can be continuously shrunk to a point without leaving the region
  • Example: Electric field $\mathbf{E}$ in a charge-free region is conservative, as its curl is zero ($\nabla \times \mathbf{E} = \mathbf{0}$) and the region is simply connected