A few questions about inverse operations.

[Isaac0427]

Hi all!
I know that the integral is the inverse of the derivative, but what about special derivative operators? What functions would undo the gradient, divergence and curl? And what about special integrals, such as line and surface integrals? Are there different derivatives/integrals that are defined as the inverse of these special calculus operators? Would they be of any practical use?

Thanks!

 

[Charles Link]

The inverse function to recover the function from the gradient operation e.g. $\nabla F$ is $F=\int \nabla F \cdot d \vec{s}$ where $d \vec{s}=dx \hat{i} +dy \hat{j} +dz \hat{k}$. These are three separate integrals (x,y, and z) that are often easily performed... The $\nabla \cdot \vec{E}$ has an integral solution. (There also can be homogeneous solutions as well to both this one and the integral solution to the curl that needed to be added to this particular integral solution). In MKS let's assume $\nabla \cdot \vec{E}(x)=\rho (x) /\epsilon_o$. (It works for other vector functions as well besides the electric field $\vec{E}$). Then $\vec{E}(\vec{x})=\int [(1/(4 \pi \epsilon_o)) \rho(\vec{x}') (\vec{x}-\vec{x}')/|\vec{x}-\vec{x}'|^3] d^3 \vec{x}'$ (basically the inverse square law for the electric charge distribution $\rho (\vec{x})$)... The integral solution to the curl takes the Biot-Savart form. e.g. For $\nabla \times \vec{B}(\vec{x})=\mu_o \vec{J}(\vec{x})$, the solution of this is the Biot-Savart integral form: $B(\vec{x})=\int [(1/(4 \pi \mu_o)) \vec{J}(\vec{x}') \times (\vec{x}-\vec{x}')/|\vec{x}-\vec{x}'|^3] d^3 \vec{x}'$. Oftentimes this curl integral needs a homogeneous solution of $\nabla \times \vec{B}=0$ added to it. (I don't think this is usually the case with the B field, but for the H field and in other cases it often requires a homogeneous solution as well) And you ask, are they of any practical use? The answer is yes. The integral solution to the divergence equation is often used in E&M calculations. The Biot-Savart equation is well known, but it doesn't seem to be emphasized so much as the integral solution to the curl equation in the E&M textbooks. For the curl B equation, the integral solution that is more often shown is Ampere's law in integral form (for the steady state case) which comes from Stokes theorem: $\oint \vec{B} \cdot d \vec{l}=\mu_o I$.

 

[Charles Link]

To be more complete, one or two additional items need to be added to the above response. It's not exactly an inverse function, but for the $\nabla \cdot \vec{E}=\rho/\epsilon_o$, there is a very standard solution used in E&M that even allows a solution for $\vec{E}$ in problems with sufficient symmetry. The "solution" is called Gauss's law: $\int \nabla \cdot {E} \ d^3 \vec{x}=\int \vec{E}\cdot \hat{n}\ dA$ where $\hat{n}$ is the outward pointing unit vector normal to the surface which encloses the volume in the volume integral. Thereby the "flux" of the electric field over any volume is $Q/\epsilon_o$ where $Q$ is the enclosed charge. (With sufficient symmetry $\int \vec{E} \cdot \hat{n} \ dA =EA \$ ,etc.). This bit of mathematics is used extensively in E&M calculations-much more so than even the inverse square integral solution given in post #2. For the curl, there is likewise a rather unique solution know as Stoke's theorem: $\int \nabla \times \vec{B} \cdot \hat{n}\ dA=\oint \vec{B} \cdot d \vec{l}$ where the line integral is the (counterclockwise) loop around the surface over which the surface integral is performed. This is also used extensively in E&M calculations in the integral form of Ampere's law that was mentioned in post #2.