Vector Calculus Review

Transcription

1 Course Instructor Dr. Ramond C. Rumpf Office: A-337 Phone: (915) Vector Calculus Review EE3321 Electromagnetic Field Theor Outline Mathematical Preliminaries Phasors, vectors, notation Math with Vectors Coordinate Sstems Notation, differentials Visualiation of Fields & Operations 1

2 Mathematical Preliminaries Phasors (1 of 2) A time-harmonic function can be written as cos t A t Recall Euler s Identit e j cos jsin This let s us write the function (t) as j t t Re Ae 2

3 Phasors (2 of 2) In linear sstems, frequenc does not change. We can epress time-harmonic signals without eplicitl using the frequenc term. This compact notation is a phasor. Y j Ae Polar Vs. Rectangular Form A phasor in polar form is written as j Y Ae or A The same phasor written in rectangular form is Y j Rect Polar 2 2 A tan 1 Polar Rect Acos Asin 3

4 Phasor Arithmetic Addition F F j Subtraction F F j Multiplication F F AA Division F F A A Scalars & Vectors Scalar Numbers Scalars contain onl one piece of information, magnitude. Scalars can be real or comple. Phasors are scalar quantities. Eamples: 7,, -1.34, etc. Vectors Vectors have both a magnitude and a direction. Eamples: Velocit, force, electromagnetic fields 4

5 Vector Notation direction Parallel to paper Out of paper Think of seeing the point of an arrow Into the paper Think of seeing the back of an arrow Note: Despite the arrow etending awa from the point, a vector is describing something at that specific point and it does not actuall etend outward. What Can Vectors Conve? Position Distance Disturbance Position relative to the origin. Vectors can indicate distance, but the origin is not given. A vector can represent a directional disturbance. Thank of this as a push. 5

6 Simple Vector Calculations 3D Vector A Aˆ Aˆ Aˆ Vector Magnitude A A A A Unit Vector Aˆ A A Math With Vectors 6

7 Visualiation of Vector Addition & Subtraction Vector Addition Vector Subtraction U V U V U V V U Vector Addition & Subtraction Cartesian Clindrical Spherical Starting Vectors U U aˆ U aˆ U aˆ V V aˆ V aˆ V aˆ Addition U V U V aˆ U V aˆ U V aˆ Subtraction U V U V aˆ U V aˆ U V aˆ Starting Vectors U U aˆ U aˆ U ˆ a V V aˆ V aˆ V aˆ Addition U V U V aˆ U V aˆ U V aˆ Subtraction U V U V aˆ U V aˆ U V aˆ Starting Vectors U U ˆ ˆ ˆ rar U a U a V V aˆ V aˆ V aˆ r r Addition U V U V aˆ r r r U V aˆ U V aˆ Subtraction U V U V aˆ r r r U V aˆ U V aˆ 7

8 The Dot Product, A B The dot product is all about projections. That is, calculating how much of one vector lies in the direction of another vector. A B AB A B cos A B A B A B Projection of A onto B B B A B A A B B 2 B B B Magnitude Direction A ˆ B B B B A A B B 8

9 Projection of B onto A A A B A B B A A 2 A A A Magnitude Direction B ˆ A A A A B A A B The Dot Product Test We can use the dot product to test of two vectors are perpendicular. If the are, the component of one along the other must be ero. AB0 when A B 9

10 The Cross Product, A B The cross product is all about area and calculating vectors that are perpendicular to and. A A B Area AB B A B AB A B sinaˆ aˆ n is a unit vector perpendicular to the plane defined b A and B. n Calculating Cross Products (1 of 2) Suppose we wish to calculate the cross product A B. A Aˆ Aˆ Aˆ B B ˆB ˆB ˆ Step 1 Construct an augmented matri. ˆ ˆ ˆ ˆ ˆ A A A A A B B B B B First two columns are repeated outside of the matri. 10

11 Calculating Cross Products (2 of 2) Step 2 Multipl elements along the diagonals. A Bˆ A Bˆ A Bˆ Step 3 Make left-hand side products negative. A Bˆ A Bˆ A Bˆ Step 4 Add up all of the products. AB A B A B ˆ A B A B ˆ A B A B ˆ The Cross Product Test We can use the cross product to test of two vectors are parallel. If the are, the cross product will be ero because the angle between the vectors is ero. AB0 when A B 11

12 Vector Algebra Rules Commutative Laws AB BA Associative Laws Distributive Laws A B C A B A C Self-Product 2 AA A AB BA A BC AB C A B C A B A C AA0 Vector Triple Products Scalar Triple Product The scalar triple product is the volume of a parallelpiped. BC A CA B AB C Vector Triple Product The vector triple product arises when deriving the wave equation. A BC B AC C AB 12

13 Coordinate Sstems Tpes of Integrations Ordinar Line Integral d L L Ordinar Surface Integral ds S S Ordinar Volume Integral dv V V Closed-Contour Line Integral d L L Closed-Contour Surface Integral ds S S Closed-Contour Volume Integral v dv? 13

14 Cartesian Coordinates Image courtes of Wikipedia. Cartesian Differentials (1 of 2) dv ddd d d d 14

15 Cartesian Differentials (2 of 2) ds ddaˆ dv ddd d ds ddaˆ d d Clindrical Coordinates Image courtes of Wikipedia. 15

16 Clindrical Differentials (1 of 2) d d dv d dd d d d Clindrical Differentials (2 of 2) ds ddaˆ dv d dd d d d d d ds ds ddaˆ ddaˆ 16

17 Spherical Coordinates Image courtes of Wikipedia. Spherical Differentials 0 r

18 Summar of Differentials Cartesian Coordinates Clindrical Coordinates Spherical Coordinates Visualiation of Fields & Operations 18

19 Scalar Field Vs. Vector Field Scalar Field,, magnitude,, f Vector Field, v, magnitude,, direction,, Alternate Visualiation of a Vector Field Arrows conve magnitude and direction. Background color also conves magnitude. 19

20 Isocontour Lines Isocontour lines trace the paths of equal value. Closel space isocontours conves that the function is varing rapidl. Gradient of a Scalar Field (1 of 3) We start with a scalar field f, 20

21 Gradient of a Scalar Field (2 of 3) then plot the gradient on top of it. Color in background is the original scalar field. f, Gradient of a Scalar Field (3 of 3) The gradient will alwas be perpendicular to the isocontour lines. 21

22 Divergence of a Vector Field (1 of 2) Suppose we start with the following vector field A, Divergence of a Vector Field (2 of 2) We then plot the divergence as the color in the background. The arrows are the original vector function. A, 22

23 Curl of a Vector Field (1 of 2) Suppose we start with the following vector field B, Curl of a Vector Field (2 of 2) The color in the background is the magnitude of the curl. The direction is either into, or out of, the screen. Red indicates + direction while blue indicates direction. B, 23

24 Summar of Vector Operations Operation Input Output Vector Addition & Subtraction Vectors Vectors U V Dot Product U V Vectors Scalar Cross Product UV Vectors Vector Gradient f Scalar Function Vector Function Divergence U Vector Function Scalar Function Curl U Vector Function Vector Function 24