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Math for Physics

Curl

The curl of a vector field is a measure of the field's tendency to circulate about a point. James Clerk Maxwell invented the term "curl". Curl is a point function Circulation Circulation (the figure below) is defined as a line integral of the vector field A over a closed path C: Curl Curl at a specific point is defined as the circulation per unit area over an infinitesimal path around that point. where C is a path around the point of interest and the ΔS is the surface area enclosed by that path. Cur is a vector quantity and its direction is the normal...

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Surface Integral

Surface integral is the area integral of a scalar function or vector field over a specified surface. This is best understood by looking at a practical problem shown below. The Problem: The area density (the mass per unit area) of this surface varies with x and y, we need to find the total mass of the surface. The Solution: We can divide the surface into two-dimensional segments over each of which the area density δ(x,y) is approximately constant. For each segment, the density is δi, the area is dAi, and the mass for this small segment is δi dAi. Then the total mass will be the...

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Direction Cosines

In analytic geometry, the direction cosines (also called directional cosines) of a vector are the cosines of the angles between the vector and the three coordinate axes. Equivalently, they are the contributions of each component of the basis to a unit vector in that direction.

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Orthogonal Coordinate Systems - Cartesian, Cylindrical, and Spherical

Base Vectors In a  three-dimensional space, a point can be located as the intersection of three surfaces. The three surfaces are described by u1 = constant u2 = constant u3 = constant u1, u2, and u3 need not all be lengths as shown in the table below.   u1 u2 u3 Cartesian Coordinate System x y z Cylindrical Coordinate System r φ z Spherical Coordinate System R θ φ   If these three surfaces (in fact, their normal vectors) are mutually perpendicular to each other, we call them orthogonal coordinate system.   Cartesian Coordinate System: In Cartesian coordinate system, a point...

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Vector Algebra

Why Vector Algebra? Most serious analysis of optical signal generation and propagation are based on Maxwell's Equations and boundary value problems such as optical modes and their propagation in optical fibers, waveguides, arrayed waveguide gratings (AWG), etc. Maxwell's equations talk about the relationship of electric field intensity E, magnetic field intensity B,  the displacement current density D, and their gradient, divergence, curl operations. These quantities all have both magnitude and direction and are called vectors. So one must have a solid understanding of vector analysis in order to fully understand and appreciate Maxwell's equations and optical signal propagations in optical fibers and waveguides.   Vector...

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