Electromag (Math)

SCALAR

Has magnitude only

• Distance

• Energy, J

• Specific Heat

• Speed

• Calorie

• Volume

• Current

VECTOR

Has magnitude and direction

• Displacement

• Weight, N

• Momentum

• Velocity

• Density

• Acceleration

Vector Analysis

Branch of mathematics that deals with quantities that have both magnitude and direction

Z axis

Applicate

Position Vector

• Vector pointing to a certain direction and always starts at the origin (0,0,0)

• Measured by final - initial

Unit Vectors

Vectors pointing to a certain direction and has a magnitude of 1

Magnitude of a Poisition Vector

Direction only of Position Vector

Displacement Vector

• Vector between 2 points

• The first letter is the initial point

Magnitude of Displacement Vector

Addition Vector Operation

Subtraction Vector Operation

Scalar Multiplication

• “scale” - expand or shrink

Dot Product

[Vector Multiplication]
- Also called Scalar Product

-Real number that is the product of the lengths of 2 vectors and the cosine of the angle between them

Cross Product

[Vector Multiplication]
- Also called Vector Product

Projection

Trying to find a part of a vector that lies in the direction of another vector

Scalar Projection

Projection with Magnitude only

Vector Projection

Projection with Magnitude and direction

Magnitude of the Vector/Cross Product

• This is the area of parallelogram

Rule for Dot Product

Direction for Vector/Cross Product

Vector/Cross Product Formula

Right Hand Screw Rule

Pointer Finger: 1st Vector
Middle Finger: 2nd Vector
Thumb: Direction
example: i = j (Pointer) x k (Middle finger)

Right Hand Screw Rule Position of Finger

+i - away from you
+j - to the left
+k - above you

Volume of a Parallelepiped (Triple Scalar Product)

Shortcut: Just solve for the determinant of the 3 vectors

“Orthogonal”

Means “Perpendicular”

If Vector A orthogonal with Vector B, then

What will make the cross product equal to zero

A, B, C are coplanar (they are on the same plane)

if the triple scalar product is equal to 0 then

Multivariable Calculus

Vector Calculus is also called as

Vector Calculus

Branch of mathematics that is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidian space

Differentiation of a Vector

• Derive per component

Multiplication Rule in Differentiation of a Vector

Rule for Vector Product

Integration of a Vector

Integrate per term

Scalar Field

Function of a spatial coordinates giving a single scalar value at every point (x,y,z)

Operator of the Gradient of a scalar

nabla / del operator

Gradient of a scalar

• Vector

• Sleepest ascent

• Toward upward

• 1st derivative of the scalar field

Equivalent of nabla / del

Solenoidal Field

What do you call a field with zero or “vanishing” divergence

Divergence of a Vector

• Scalar

Laplacian of a Vector

• 1nd derivative of a scalar field

• Divergence of the gradient

Irrotational / Conservative field

If the curl of the field is zero then it is a

Curl of a Vector

Tendency of a vector field to rotate about a point

0

divergence of a curl

0

curl of a gradient

Laplacian Operator

nabla square

Ampere’s Circuital Law

1st Maxwell’s Law

Gauss’s Law