ELEC 3600 T UTORIAL 2 V ECTOR C ALCULUS Alwin Tam Rm. 3121A

W HAT H AVE W E L EARNT S O F AR ? 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

S CALAR A ND V ECTOR F IELD 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

S CALAR A ND V ECTOR F IELD (C ONT.) Is temperature a scalar quantity? A. Yes B. No Answer: A, because it can be completely described by a number when someone ask how hot is today. Is acceleration a scalar quantity? A. Yes B. 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.

V ECTOR C ALCULUS 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 line integral)

D IFFERENTIAL L ENGTH, V OLUME AND S URFACE (C ARTESIAN C OORDINATE ) 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 Differential is infinitely small difference between 2 quantities

D IFFERENTIAL L ENGTH, V OLUME AND S URFACE (C YLINDRICAL C OORDINATE ) All vector components MUST have spatial units i.e. meters, cm, inch etc.

D IFFERENTIAL L ENGTH, V OLUME A ND S URFACE (S PHERICAL C OORDINATE ) z x y All vector components MUST have spatial units i.e. meters, cm, inch etc.

L INE I NTEGRAL 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 Line integral Definite integral Diagram Maths description Result Area under the curve A measure of the total effect of a given field along a given pathfield Information required 1.Vector field expression A 2.Path expression 1.Function f(x) 2.Integral limits Integral limits depends on path

S URFACE & V OLUME I NTEGRAL 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 scalars involve inside the integral Result of volume integral is a scalar

S URFACE & V OLUME I NTEGRAL (C ONT.) Surface integral Volume integral Diagram Maths description Result 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 field passing through a given surface Information required 1.Vector field expression A 2.Surface expression 1.Scalar Function  v 2.Volume expression Integral limits depends on surface Integral limits depends on volume

P ROBLEM 1 Given that, calculate the circulation of F around the (closed) path shown in the following figure.

D EL O PERATOR Vector differential operator Must operate on a quantity (i.e. function or vector) to have a meaning Mathematical form: Cartesian CylindricalSpherical

S UMMARY O F G RAD, D IV & C URL GradientDivergenceCurl Scalar f(x,y) Vector A Expression (Cartesian) Expression (Cylindrical) Expression (Spherical) ResultVector ScalarVector

S UMMARY O F G RAD, D IV & C URL GradientDivergenceCurl Physical meaning A vector that gives direction of the maximum rate of change of a quantity i.e. temp i.e. Flux out < flux in i.e. Flux out > flux in Incompressible Flux out = flux in

D IVERGENCE T HEOREM Divergence theorem: Total outward flux of a vector field A through a closed surface S is the same as the volume integral of div A. i.e. Transformation of volume integral involving div A to surface integral involving A Equation: Physical meaning: The total flux from field A passing through a volume V is equivalent to summing all the flux at the surface of V.

P ROBLEM 2 (M IDTERM E XAM 2013) Verify the divergence theorem for the vector r 2 a r within the semisphere.

S TOKE ’ S T HEOREM Stoke’s Theorem: The line integral of field A at 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 curl A 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.

L APLACIAN O F A S CALAR F UNCTION 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!!

P ROBLEM 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? 1 2 3