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Maxwell's equations can directly give inhomogeneous wave equations for the electric field E and magnetic field B. [1] Substituting Gauss's law for electricity and Ampère's law into the curl of Faraday's law of induction, and using the curl of the curl identity ∇ × (∇ × X) = ∇(∇ ⋅ X) − ∇ 2 X (The last term in the right side is the vector Laplacian, not Laplacian applied on ...
Duhamel's principle is the result that the solution to an inhomogeneous, linear, partial differential equation can be solved by first finding the solution for a step input, and then superposing using Duhamel's integral. Suppose we have a constant coefficient, m-th order inhomogeneous ordinary differential equation.
The inhomogeneous Helmholtz equation is the equation + = , where ƒ : R n → C is a function with compact support, and n = 1, 2, 3. This equation is very similar to the screened Poisson equation , and would be identical if the plus sign (in front of the k term) were switched to a minus sign.
Following are some of the several other mathematical formalisms of Maxwell's equations, with the columns separating the two homogeneous Maxwell equations from the two inhomogeneous ones. Each formulation has versions directly in terms of the electric and magnetic fields, and indirectly in terms of the electrical potential φ and the vector ...
The inhomogeneous Fredholm integral equation = + (,) ()may be written formally as = which has the formal solution =. A solution of this form is referred to as the resolvent formalism, where the resolvent is defined as the operator
The inhomogeneous Cauchy–Riemann equations consist of the two equations for a pair of unknown functions u(x, y) and v(x, y) of two real variables = (,) + = (,) for some given functions α( x , y ) and β( x , y ) defined in an open subset of R 2 .
These representations are for a homogeneous medium, an approximation in an inhomogeneous medium. A matrix representation for an inhomogeneous medium was presented using a pair of matrix equations. [1] A single equation using 4 × 4 matrices is necessary and sufficient for any homogeneous medium.
In vacuum (or for the microscopic equations, not including macroscopic material descriptions), Maxwell's equations can be written as two tensor equations. The two inhomogeneous Maxwell's equations, Gauss's Law and Ampère's law (with Maxwell's correction) combine into (with (+ − − −) metric): [3]