The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge-Ampère measures

The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write...

The two theorems developed by Pappus and Guldinus are widely used in mathematics, engineering, and physics to find the surface area and volume of any body of revolution. This is done by revolving a plane curve around an axis that does not intersect the curve to find its surface area or revolving a plane area around a non-intersecting axis to calculate its volume.
For finding the surface area, consider a differential line element that generates a ring with surface area dA when revolved.

The Moment-Area Theorem is crucial in structural engineering for analyzing beam bending, particularly in applications like building floor supports. This theorem utilizes the geometric properties of the elastic curve, which depicts how a beam deforms under load, to simplify the calculations of deflections and slopes.
The theorem is divided into two parts. The first part connects the angle between tangents at any two points on the beam's elastic curve to the area under a curve derived by...

The principle of Saint-Venant postulates that the stress distribution within a structural member does not rely on the precise method of load application except in the vicinity of the load application points. Consider a scenario where loads are centrally applied on two plates. In this case, the plates move toward each other without any rotation. This movement causes the member to contract in length and expand in width and thickness. Uniform deformation across all elements and maintaining...

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...

Castigliano's theorem analyzes displacements and rotations in elastic structures. It relates the derivative of elastic strain energy to the applied forces or moments, allowing for the calculation of deformations. The theorem states that the partial derivative of the total strain energy of a system with respect to a specific load results in the displacement at the point where the load is applied. This principle applies to both forces and moments.