Electromagnetic Theory Prof. Michael J. Ruiz, UNC Asheville (doctorphys on YouTube) Chapter A Notes. Vector Analysis

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1 Electromagnetic Theory Prof. Michael J. Ruiz, UNC Asheville (doctorphys on YouTube) Chapter A Notes. Vector Analysis A0. Vector Figure Courtesy OpenStax College. Vector Addition and Subtraction: Analytical Methods, Connexions Website. June 20, 202. A = A i+ A x y A x = A i and Ay = Ay x A. Vector Addition and Subtraction Figures Courtesy OpenStax College A = Ax i+ Ay B = Bx i+ By R = ( A + B ) i+ ( A + B ) x x y y

2 The Resultant: R = A + B = ( A + B ) i+ ( A + B ) = R i+ R x x y y x y Adding the Negative Vector (Subtraction): A B = ( A B ) i+ ( A B ) x x y y Image Courtesy Acdx, Wiipedia Examples of vectors in three dimensions. a = a i+ a + a A = A i+ A + A F = F i+ F + F A2. Scalar Multiplication A3. Dot Product α A = α ( A i+ A + A ) α A = α A i+ α A + α A Image from Wiimedia Commons A B = AB cosθ Note: i i = = = ()() cos 0 = i = i = = ()() cos 90 = 0

3 Also note B A = BAcosθ = AB cosθ = A B Using the rules for the dot product of the unit vectors we arrive at the following. A B = ( A i+ A + A ) ( B i+ B + B ) A B = A B i i+ A B i + A B i x x x y x z + A B i+ A B + A B y x y y y z + A B i+ A B + A B z x z y z z A B = A B + A B + A B x x y y z z e More notation: = i, e2 =, and e3 =. Then A = A e + A2 e2 + A3 e3. The unit vectors are also called basis vectors and we use subscripts that can tae on values, 2, and 3. The dot product of two arbitrary unit vectors can then be written as e e = δ where δ = if i = and δ = 0 if i i i i i. Here is more new notation: A 3 = i= A e i i and B 3 = i= B e i i A B = A e B e = A B e e = A B δ = A B i i i i i i i i i= = i= = i= = i= A B = A B + A B + A B x x y y z z A B = A B + A B + A B or

A = A e Einstein Summation Convention: i i B = B e and i i A B = A e B e = A B e e = A B δ = A B i i i i i i i i Leopold Kronecer (823 823-89 89) Courtesy School of Mathematics and Statistics

Find the angle theta between the two vectors using the two dot product definitions. Chec your answer with a graphical diagram. A4. Cross Product A = i and B = i+.

4 A = A e Einstein Summation Convention: i i B = B e and i i A B = A e B e = A B e e = A B δ = A B i i i i i i i i Leopold Kronecer ( ) Courtesy School of Mathematics and Statistics University of St. Andrews, Scotland The Kronecer Delta symbol is defined as and named after the German mathematician Leopold Kronecer. It is a symmetric symbol. PA (Practice Problem). Find the angle theta between the two vectors using the two dot product definitions. Chec your answer with a graphical diagram. A4. Cross Product A = i and B = i+. Cross Product (Images Courtesy Acdx, Wiipedia) A B = AB sinθ n where the unit vector n, a b = absinθ n is perpendicular to the plane formed by a and b, according to the righthand rule as shown in the lower figure. Or you can use the "right-hand screwdriver rule" where you get under the plane and apply the screwdriver to turn "a" into "b" advancing along "n". By the way ab sinθ is the area shown in the parallelogram.

5 Image Courtesy Acdx, Wiipedia Note that if you flip the order of the vectors, you get a vector in the opposite direction according to the right-hand rule. b a = a b b a = ba sin θ ( n) and B A = A B The right hand-rule with the unit vectors gives us these relations below. Image Courtesy Acdx, Wiipedia i = i = i i = 0 = i = i = 0 i = i = = 0 We now apply these rules. A B = ( Ax i+ Ay + Az ) ( Bx i+ By + Bz ) A B = A B i i+ A B i + A B i x x x y x z y x y y y z wors out to: + A B i+ A B + A B + Az Bx i+ Az By + Az Bz A B = A B 0 + A B + A B ( ) x x x y x z + A B ( ) + A B 0 + A B i y x y y y z + A B + A B ( i) + A B 0 z x z y z z

6 A B = ( A B A B ) i+ ( A B A B ) + ( A B A B ) y z z y z x x z x y y x i A B = A A A B B B = i( A B A B ) ( A B A B ) + ( A B A B ) y z z y x z z x x y y x We now switch to our index notation. where e = i, e2 =, and e3 =. The cross-product rules can be summarized by writing ei e ε i e = where Tullio Levi-Civita (873-94) Courtesy School of Mathematics and Statistics University of St. Andrews, Scotland The symbol ε i is called the Levi-Civita or permutation symbol. It is an antisymmetric symbol. If you swap any two indices you introduce a minus sign. If any two indices are the same you get zero. A B = A e B e = A B e e i i i i i= = i= = 3 3 A B A B ε e = i= = i i

The same with Einstein's summation convention is: A B = A e B e = A B e i i i εi PA2 (Practice Problem). Find the angle theta between the two vectors using the two cross product definitions.

7 The same with Einstein's summation convention is: A B = A e B e = A B e i i i εi PA2 (Practice Problem). Find the angle theta between the two vectors using the two cross product definitions. Chec your answer against PA. A5. Tensors A = i and B = i+ Tensor of Ran 0. This is your scalar. A single number is all you need. There is no directional vector or anything lie that. The length of a vector stripped of its direction is a scalar. Another example is temperature at each point in a room: T = T(x,y,z) or you can add the time variable so the temperatures change in time. Below is a snapshot of the temperatures across the United States at the time of the writing of this chapter.. Courtesy The Weather Channel

Tensor of Ran. This is your vector. It has magnitude and direction. It can also be a function of the spatial coordinates as well as time. Courtesy Weather Underground, Inc.

8 Tensor of Ran. This is your vector. It has magnitude and direction. It can also be a function of the spatial coordinates as well as time. Courtesy Weather Underground, Inc. Wind velocity has magnitude (the speed) and direction. The length of the vector arrows indicate the magnitude of the velocity and the arrow points in the direction of the wind. Technically, speed is a scalar, the magnitude. When you promote speed to a vector you add the direction. However, often velocity is used informally for ust speed. Charge Image Courtesy Tony Wayne Here is a vector field produced by a plus charge. Note the symmetry as all vectors points outward away from the positive charge. Also note that the lengths of the vectors decrease as you get farther away from the charge. The strength weaens according to the inverse square law. In contrast to the weather case this field has a simple formula.

Tensor of Ran 2. Among friends, you can thin of a tensor of ran 2 as needing 3 x 3 = 9 components. Courtesy Sanpaz, Wiipedia The stress tensor is an example.

9 Tensor of Ran 2. Among friends, you can thin of a tensor of ran 2 as needing 3 x 3 = 9 components. Courtesy Sanpaz, Wiipedia The stress tensor is an example. We need to consider the force on each of the three main faces defined by the three unit vectors. On each surface there is a normal force and two shear (sideway) forces. We need not consider all 6 faces since mechanical equilibrium guarantees that there will be opposing forces and torques on the opposite sides. This means we need 9 quantities to define the stress. Matrix notation will assist us here. For the tensors of Ran 0,, and 2 respectively, we can write for three dimensional space. s = [ T ] = T A x 2 3 A = A y i A z = For two dimensions we have for tensors of Ran 0,, and 2 respectively listed below. s = [ T ] = T Ax A = A y M i M M 2 = M 2 M 22 But not all matrices are tensors. There are transformation properties that need to be satisfied. However, if you need vector components as in the stress analysis, then you are on good grounds that you are probably dealing with a tensor. Tensor of Ran 3. What would this be? How about a Tensor of Ran n in m dimensions?

Vectors for Physics. AP Physics C

Vectors for Physics. AP Physics C Vectors for Physics AP Physics C A Vector is a quantity that has a magnitude (size) AND a direction. can be in one-dimension, two-dimensions, or even three-dimensions can be represented using a magnitude