ELEC 3600 Tutorial 2 Vector Calculus

1. ELEC 3600 Tutorial 2 Vector Calculus Alwin Tam amwtam@ust.hk Rm. 3121A

2. What Have We Learnt So Far? • Classification of vector & scalar fields • Differential length, area and volume • Line, surface and volume integrals • Del operator • Gradient of a scalar • Divergence of a vector • Divergence theorem • Curl of a vector • Stokes’ theorem • Laplacian of a scalar

3. Scalar And Vector Field • What is scalar field? • Quantities that can be completely described from its magnitude and phase. i.e. weight, distance, speed, voltage, impedance, current, energy • What is a vector field? • Quantities that can be completely described from its magnitude, phase and LOCATION. i.e. force, displacement, velocity, electric field, magnetic field • Need some sense of direction i.e. up, down right and left to specify

4. Scalar And Vector Field (Cont.) • Is temperature a scalar quantity? • Yes • No Answer: A, because it can be completely described by a number when someone ask how hot is today. • Is acceleration a scalar quantity? • Yes • No Answer: B, because it requires both magnitude and some sense of direction to describe i.e. is it accelerating upward, downwards, left or right etc.

5. Vector Calculus • What is vector calculus? • Concern with vector differentiation and line, surface and volume integral • So why do we need vector calculus?? • To understand how the vector quantities i.e. electric field, changes in space (vector differential) • To determine the energy require for an object to travel from one place to another through a complicated path under a field that could be spatially varying (line integral) i.e. • To pass ELEC 3600!! (vector differential and lineintegral) W =

6. Differential Length, Volume and Surface (Cartesian Coordinate) Differential is infinitely small difference between 2 quantities • Differential length • A vector whose magnitude is close to zero i.e. dx, dy and dz → 0 • Differential volume • An object whose volume approaches zero i.e. dv = dxdydz→ 0 (scalar) • Differential surface • A vector whose direction is pointing normal to its surface area • Its surface area |dS| approach zero i.e. shaded area ~ 0 • Calculated by cross product of two differential vector component

7. Differential Length, Volume and Surface (Cylindrical Coordinate) All vector components MUST have spatial units i.e. meters, cm, inch etc.

8. Differential Length, Volume And Surface (Spherical Coordinate) z All vector components MUST have spatial units i.e. meters, cm, inch etc. y x

9. Line Integral • Line integral: Integral of the tangential component of vector field A along curve L. • 2 vectors are involve inside the integral • Result from line integral is a scalar Definite integral Line integral Diagram Maths description A measure of the total effect of a given field along a given path Area under the curve Result Information required Vector field expression A Path expression Function f(x) Integral limits Integral limits depends on path

10. Surface & Volume Integral • Surface integral: Integral of the normal component of vector field A along curve L. • Two vectors involve inside the integral • Result of surface integral is a scalar • Volume integral: Integral of a function f i.e. inside a given volume V. • Two scalarsinvolve inside the integral • Result of volume integral is a scalar

11. Surface & Volume Integral (Cont.) Volume integral Surface integral Diagram Maths description A measure of the total effect of a scalar function i.e. temperature, inside a given volume A measure of the total flux from vector fieldpassing through a given surface Result Information required Vector field expression A Surface expression Scalar Function rv Volume expression Integral limits depends on volume Integral limits depends on surface

12. Problem 1 • Given that , calculate the circulation of F around the (closed) path shown in the following figure. Solution:

13. Del Operator • Vector differential operator • Must operate on a quantity (i.e. function or vector) to have a meaning • Mathematical form: Cylindrical Spherical Cartesian

14. Summary Of Grad, Div & Curl Gradient Divergence Curl must operate on Scalar f(x,y) Vector A Vector A Expression (Cartesian) Expression (Cylindrical) Expression (Spherical) Scalar Vector Result Vector

15. Summary Of Grad, Div & Curl Gradient Divergence Curl A vector operator that describes the rotation/ununiformity of a vector field RHC rotation LHC rotation Irrotational A scalar that measures the magnitude of a source or sink at a given point SinkSource A vector that gives direction of the maximum rate of change of a quantity i.e. temp Physical meaning i.e. Flux out < flux in i.e. Flux out > flux in Incompressible Flux out = flux in

16. Divergence Theorem • Divergence theorem: • Total outward flux of a vector field A through a closed surface S is the same as the volume integral of divA. i.e.Transformation of volumeintegral involving divA to surfaceintegral involving A • Equation: • Physical meaning: The total flux from field Apassing through a volume V is equivalent to summing all the flux at the surface of V.

17. Problem 2 (Midterm Exam 2013) Verify the divergence theorem for the vector r2arwithin the semisphere.

18. Stoke’s Theorem • Stoke’s Theorem: • The line integral of field Aat the boundary of a closed surface S is the same as the total rotation of field A at the surface. i.e. Transformation of surface integral involving curlA to line integral of A • Equation: • Physical meaning: The total effect of field A along a closed path is equivalent to summing all the rotational component of the field inside the surface of which the path enclose.

19. Laplacian Of A Scalar Function • U is a scalar function of x, y, z (i.e. temperature) • Laplacian of a scalar = Divergence of a Gradient of scalar function. • Important operator when working with MAXWELL’S EQUATION!!

20. Problem 3 Given that , find (a) Where L is shown in the following figure (b) Where S is the area bounded by L (c) Is Stokes’s theorem satisfied? 2 1 3