EMLAB 1 Chapter 1. Vector analysis

EMLAB 2 Mathematics -Glossary Scalar : a quantity defined by one number (eg. Temperature, mass, density, voltage,... ) Vector : a quantity defined by a set of numbers. It can be represented by a magnitude and a direction. (velocity, acceleration, …) Field : a scalar or vector as a function of a position in the space. (scalar field, vector field, …) Scalar field Vector field

EMLAB 3 Scalar field Vector field

EMLAB 4 Vector algebra Unit vector : ‘^’ means the unit vector. Components of a vector

EMLAB 5 Products of vectors scalar product : the result of a scalar product is a scalar. vector product : the result of a vector product is a vector which is normal to the original vectors. The magnitude of C is the area of a parallelogram made by two vectors A and B.

EMLAB 6 Example of a scalar product

EMLAB 7 Calculation of vector products (+) (-) (+) (-) (+) If rotated in the direction of x,y,z, the sign of the result is (+). If rotated in the opposite direction, the sign is (-). The same rule applied to other coordinate systems with change of unit vectors. (+) (-) (+) (-)

EMLAB 8 Example of vector product torque Angular acceleration is proportional to the applied torque. Torque is proportional to the product of radius and force. If the sum of torques due to A and B has nonzero value, the seesaw is rotated.

EMLAB 9 Vector identity The order of operators and can be interchanged. The scalar triple product is equal to the volume of the parallelogram defined by the three vectors. Scalar triple product The result is the same despite the change of the order of and x interchanged.

EMLAB 10 Quantities represented by vectors Position vector : a position can be represented by a vector whose origin is specified by O. Field vector : A vector that represents physical quantities at the position specified by a position vector. Position vector Field vector Origin

EMLAB 11 O Origin Position vector Example : Velocity Velocity vector

EMLAB 12 Coordinate system Position vector : A position can be specified with a vector whose origin is at a point O. O Observation position is written by R. Source position is represented by a primed vector. 1.A coordinate system is only an apparatus that describes positions and physical phenomena. 2.Physical laws are independent of coordinate systems adopted by any observers.

EMLAB 13 We can choose a set of unit vectors which fits a specific problem. Some orthogonal coordinate systems and their unit vectors constant unit vectors The directions of two unit vectors change with observation positions.

EMLAB 14 O O O Position vectors

EMLAB 15 Field vectors

EMLAB 16 For cylindrical coordinate systems, unit vectors ρ and φ are functions of an observer’s position.

EMLAB 17 For spherical coordinate systems, unit vectors r, θ, and φ are functions of a position.

EMLAB 18 Circular cylindrical coordinate Cylindrical coordinate

EMLAB 19 Example D1.5. Represent the position D(x= -3.1, y= 2.6, z= -3) using circular cylindrical coordinate system.

EMLAB 20 Example For an observation point P(1,2,3) in rectangular coordinate, express the position vector and E-field vector using unit vectors of cylindrical coordinate. (1) Position vector (2) Field vector

EMLAB 21 Spherical coordinate

EMLAB 22 Example For an observation point P(1,2,3) in rectangular coordinate, express the position vector and E-field vector using unit vectors of spherical coordinate. (1) Position vector (2) Field vector

EMLAB 23 O Origin Position vector changed Displacement vector

EMLAB 24 Rectangular coordinate If the position can be described by line segments, rectangular coordinate systems are convenient. Unit vectors are parallel to x, y, z axes. Their directions are fixed. (The unit vectors are constant ones.) Position vector changed Displacement vector O Origin

EMLAB 25 Cylindrical coordinate The direction of ρ is defined as that points away from z-axis. Φ-direction is orthogonal to ρ.

EMLAB 26 Unit circle : the magnitude of unit vectors are unity.

EMLAB 27 Spherical coordinate R : points away from the origin. θ : longitudinal direction Φ: latitude direction

EMLAB 28 Unit circle : the magnitude of unit vectors are unity.

EMLAB 29 Line integrals Line integral

EMLAB 30 Normal vector to the surface : a unit vector normal to the integration surface Surface integral

EMLAB 31 The coordinate of the center of the cube is (x,y,z) Volume integral differential volume in rectangular coordinate Density

EMLAB 32 To represent arbitrary physical vector quantities, the number of unit vectors in a rectangular coordinate is three. The unit vectors are parallel to x, y, z axes and the directions are constant. Rectangular coordinate Line integral Surface integral Volume integral

EMLAB 33 Line integral Example

EMLAB 34 Cylindrical coordinate 1.The unit vectors for a cylindrical coordinate are rho, phi, z. 2.The unit vector z is constant vector. 3.The directions of rho and phi are changing with positions. Line integral Surface integral Volume integral

EMLAB 35 Example Evaluate a line integral which has an integrand of F∙dr.

EMLAB 36 Spherical coordinate Unit vectors for a spherical coordinate are r, theta, phi. The directions of them all change with positions. Line integral Surface integral Volume integral

EMLAB 37 Example θ : longitudinal direction

EMLAB 38 Summary The unit vectors for a rectangular coordinate system are constant. For spherical or cylindrical coordinates, the direction of unit vectors change with position. Cylindrical coordinate Spherical coordinate Rectangular coordinate

EMLAB 39 Coordinate transformation Cylindrical coordinate Coordinate transformation matrix A is an orthogonal matrix. If a matrix is orthogonal, A -1 is equal to A T. Spherical coordinate

EMLAB 40 Integral of vector function