An existence theorem for fractional q-difference inclusions with nonlocal substrip type boundary conditions

Evaluating Areas Under Curves with DiscontinuitiesA definite integral is considered improper when the integrand is discontinuous at one of the limits of integration. This occurs when the function is undefined or becomes infinite at an endpoint, making the corresponding region under the curve unbounded. Such behavior is commonly associated with vertical asymptotes at the boundary of the interval. To properly define and evaluate these integrals, a limiting process is used to determine whether a...

When evaluating a definite integral whose integrand matches the structure of a composite function, the substitution method provides an efficient way to simplify the calculation. This method is based on reversing the chain rule from differentiation, allowing a complicated expression to be rewritten in a simpler form. When the integrand contains an inner function and its derivative, substitution naturally reduces the complexity of the problem.The core idea of substitution for definite integrals...

Consider a region consisting of several individual conductors with a definite charge density in the region between these conductors. The second uniqueness theorem states that if the total charge on each conductor and the charge density in the in-between region are known, then the electric field can be uniquely determined.
In contrast, consider that the electric field is non-unique and apply Gauss's law in divergence form in the region between the conductors and the integral form to the surface...

Current density becomes discontinuous across an interface of materials with different electrical conductivities. The normal component of the current density is continuous across the boundary.

When an electric field passes from one homogeneous medium to another, crossing the boundary between the two mediums imparts a discontinuity in the electric field. This results in electrostatic boundary conditions that depend on the type of mediums the field propagates through.
Consider a case where both the mediums across a boundary are two different dielectric materials. Recall that the electric field and electric displacement are proportional and related through the material's permittivity....

An electric field suffers a discontinuity at a surface charge. Similarly, a magnetic field is discontinuous at a surface current. The perpendicular component of a magnetic field is continuous across the interface of two magnetic mediums. In contrast, its parallel component, perpendicular to the current, is discontinuous by the amount equal to the product of the vacuum permeability and the surface current. Like the scalar potential in electrostatics, the vector potential is also continuous...