# 4: Vector Analysis

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A vector is a mathematical object that has both a scalar part (i.e., a magnitude and possibly a phase), as well as a direction. Many physical quantities are best described as vectors. For example, the rate of movement through space can be described as speed; i.e., as a scalar having SI base units of m/s. However, this quantity is more completely described as velocity; i.e., as a vector whose scalar part is speed and direction indicates the direction of movement. Similarly, force is a vector whose scalar part indicates magnitude (SI base units of N), and direction indicates the direction in which the force is applied. Electric and magnetic fields are also best described as vectors.

• 4.1: Vector Arithmetic
In mathematical notation, a real-valued vector A is said to have a magnitude A=|A| and direction a^ such that A=Aa^(4.1.1) where a^ is a unit vector (i.e., a real-valued vector having magnitude equal to one) having the same direction as A . If a vector is complex-valued, then A is similarly complex-valued
• 4.2: Cartesian Coordinates
Concepts described in that section – i.e., the dot product and cross product – are described in terms of the Cartesian system. In this section, we identify some additional features of this system that are useful in subsequent work and also set the stage for alternative systems; namely the cylindrical and spherical coordinate systems.
• 4.3: Cylindrical Coordinates
The cylindrical system is defined with respect to the Cartesian system. In lieu of x and y , the cylindrical system uses ρ , the distance measured from the closest point on the z axis,1 and ϕ , the angle measured in a plane of constant z , beginning at the +x axis ( ϕ=0 ) with ϕ increasing toward the +y direction.
• 4.4: Spherical Coordinates
The spherical system uses r , the distance measured from the origin;1 θ , the angle measured from the +z axis toward the z=0 plane; and ϕ , the angle measured in a plane of constant z , identical to ϕ in the cylindrical system.
The gradient of a scalar field is a vector that points in the direction in which the field is most rapidly increasing, with the scalar part equal to the rate of change.
• 4.6: Divergence
In this section, we present the divergence operator, which provides a way to calculate the flux associated with a point in space.
• 4.7: Divergence Theorem
The Divergence Theorem relates an integral over a volume to an integral over the surface bounding that volume. This is useful in a number of situations that arise in electromagnetic analysis. In this section, we derive this theorem.
• 4.8: Curl
Curl is an operation, which when applied to a vector field, quantifies the circulation of that field. The concept of circulation has several applications in electromagnetics.
• 4.9: Stokes' Theorem
Stokes’ Theorem relates an integral over an open surface to an integral over the curve bounding that surface. This relationship has a number of applications in electromagnetic theory.
• 4.10: The Laplacian Operator
The Laplacian relates the electric potential (i.e., V , units of V) to electric charge density (i.e., ρv , units of C/m 3 ) via a relationship known as Poisson’s Equation.

## Contributors and Attributions

This page titled 4: Vector Analysis is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Steven W. Ellingson (Virginia Tech Libraries' Open Education Initiative) .