Lecture 10: Vector Calculus II 1. Key points Vector fields Field Lines/Flow Lines Divergence Curl Maple commands VectorCalculus[Divergence] VectorCalculus[Curl] Student[VectorCalculus][FlowLine] Physics[Vector] package 2. Vector Fields and Field Lines (Flow Lines) A vector field has a vector value at each point of space and expressed as a vector-valued function. In the cartesican coordinate, it is written as Vector fields in physics: +force field ( ), +electric field ( ), +magnetic field ( ), +vector potential ( ), +current ( ). The gradient of a scalar field is a vector field. A field line is the curve where the field at every point on the curve is tangent to the curve. Example field line
2 y 1 0 1 2 x Example - Visualizing vector fields
Maple vector field tutor
Exercise Visualize the force field of a 3D simple harmonic oscillator: also the force field in the xy plane (z0). Answer. Plot
3. Divergence The dotproduct between a del operator and a vector field is called the divergence of the vector field div ( ) Since the output of dotproduct is scalar, the divergence of a vector field is a scalar field. Divergence in physics: Maxwell's equations: and. continuity equation:
0 Example - Visualizing divergence For simplicity, we consider two dimentional space first. Note that the divergence is negative when the arrows become smaller along the flow and positive when the arrows grow aslong the flow.
Exercise Find the divergence of the following vector fields. (1). (2). Solution (1) r 3 (2) V, F 4. Curl The cross product of a del operator and a vector field is called the curl of the vector field. curl ( ). Since the output of cross product is vector, the curl of a vector field is another vector field. Example - Visualizing curl
The vector field shows a rotation of arrows and the curl shows vectors perpendicular to the rotation. Thr curl extracts information about rotation in vector fields. Examples in Physics Force is conservetive if. Maxwell's equations: and. Vector potential:. Exercise Calculate the curl of the following vector fileds. (1) ( ). (2).
Solution (3). (1). (2). (3). 5. Combination of grad, div, and curl (1) div+grad (Laplacian) (2) curl+grad for any scalar field. (3) div+curl F for any vector field. (4) curl+curl F grad+div F - Laplacian F (5) div (4 F) V, (4F) 4(V, F)+F (grad 4) (6) curl (4 F) V# (4F) F)+(V4) F 4 (curl F) + (grad 4) F. (7) div (F G) V, (F G) F G F G F G F G (8) curl (F G) V# (F G) (F G+F(V, G G F G(V, F) (9) grad (F G) F G)+(F G +G F)+(G F (10) div [(grad ) (grad 4)] V, (Vs# V4) 0 for any scalar fields and 4. div+grad curl+grad div+curl F (5.1) 0 0
curl+curl F grad+div F - Laplacian F 6. Maple Physics[Vectors] Package When Maple Physics Vectors Package is loaded, many common epxressions used in physics are automatically assumed. For example, you don't have to set coordinates if you use common expression. See conventions. Concerning the coordinates, the conventions are: cartesian coordinates, cylindrical coordinates, spherical coordinates cartesian unit vectors, cylindrical unit vectors, spherical unit vectors Gradient Laplacian 1 Divergence x
Curl Change of Basis (6.2) (6.3) 7. Homework Homework 1 (1) Fimd the divergence of the following vector fields by hand and confirm them with Maple. (a) (b) (2) For a vector field, show that. Homework 2 Calculate the curl of the following vector fields. Visually compare the original field and its curl. (1)
(2). Homework 3: Vector potential and mangetic field In E&M class, magnetic field can be obtained from vector potential using a formula. Find the magnetic field from the following vector potential. where and are constants. Homework 4: Central fore field If a potential field depends only on radial coordinate in the sphrical coordinates as corresponding force is given by. Show the following properties:. The (1) (2) Homework 5: Field line vs equipotential surfacce Electirc field is related to electrostatic potential by. Show that the electric field line is perpendicular to the equipotential surface where is a constant.