# Chapter 3 Summary

# Concepts

- Vector algebra governs the laws of addition, subtraction, and multiplication of vectors, and vector calculus encompasses the laws of differentiation and integration of vectors.

- In a right-handed orthogonal coordinate system, the three base vectors are mutually perpendicular to each other at any point in space, and the cyclic relations governing the cross products of the base vectors obey the right-hand rule.

- The dot product of two vectors produces a scalar, whereas the cross product of two vectors produces another vector.

- A vector expressed in a given coordinate system can be expressed in another coordinate system through the use of transformation relations linking the two coordinate systems.

- The fundamental differential functions in vector calculus are the gradient, the divergence, and the curl.

- The gradient of a scalar function is a vector whose magnitude is equal to the maximum rate of increasing change of the scalar function per unit distance, and its direction is along the direction of maximum increase.

- The divergence of a vector field is a measure of the net outward flux per unit volume through a closed surface surrounding the unit volume.

- The divergence theorem transforms the volume integral of the divergence of a vector field into a surface integral of the field’s flux through a closed surface surrounding the volume.

- The curl of a vector field is a measure of the circulation of the vector field per unit area Δ s, with the orientation of Δ s chosen such that the circulation is maximum.

- Stokes’s theorem transforms the surface integral of the curl of a vector field into a line integral of the field over a contour that bounds the surface.

- The Laplacian of a scalar function is defined as the divergence of the gradient of that function.

# Mathematical and Physical Models

Distance Between Two Points

d=[(x 2−x 1)2+(y 2−y 1)2+(z 2−z 1)2]1/2 d=[r 2 2+r 1 2−2 r 1 r 2 cos (ϕ 2−ϕ 1)+(z 2−z 1)2]1/2 d={R 2 2+R 1 2−2 R 1 R 2 [cos θ 2 cos θ 1+sin θ 1 sin θ 2 cos (ϕ 2−ϕ 1)]}1/2

Coordinate Systems Table 3-1

Coordinate Transformations Table 3-2

Vector Products

A⋅B=A B cos θ A B A×B=n ˆ A B sin θ A B A⋅(B×C)=B⋅(C×A)=C⋅(A×B)A×(B×C)=B(A⋅C)−C(A⋅B)

Divergence Theorem

∫υ∇⋅E d υ=∮S E⋅d s

Vector Operators

∇T=x ˆ∂T∂x+y ˆ∂T∂y+z ˆ∂T∂z∇⋅E=∂E x∂x+∂E y∂y+∂E z∂z∇×B=x ˆ (∂B z∂y−∂B y∂z)+y ˆ (∂B x∂z−∂B z∂x)+z ˆ (∂B y∂x−∂B x∂y)∇2 V=∂2 V∂x 2+∂2 V∂y 2+∂2 V∂z 2

(see Appendix C for cylindrical and spherical coordinates)

Stokes’s Theorem

∫S(∇×B)⋅d s=∮C B⋅d l

# Important Terms

Provide definitions or explain the meaning of the following terms:

- azimuth angle

- base vectors

- Cartesian coordinate system

- circulation of a vector

- conservative field

- cross product

- curl operator

- cylindrical coordinate system

- differential area vector

- differential length vector

- differential volume

- directional derivative

- distance vector

- divergence operator

- divergence theorem

- divergenceless

- dot product

- field lines

- flux density

- flux lines

- gradient operator

- irrotational field

- Laplacian operator

- magnitude

- orthogonal coordinate system

- position vector

- radial distance r

- range R

- right-hand rule

- scalar product

- scalar quantity

- simple product

- solenoidal field

- spherical coordinate system

- Stokes’s theorem

- vector product

- vector quantity

- unit vector

- zenith angle