Report

ELEC 3600 TUTORIAL 2 VECTOR CALCULUS Alwin Tam [email protected] Rm. 3121A 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 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 SCALAR AND VECTOR FIELD (CONT.) 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. 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. W= ∙ To pass ELEC 3600!! (vector differential and line integral) DIFFERENTIAL LENGTH, VOLUME AND SURFACE (CARTESIAN COORDINATE) Differential is infinitely small difference between 2 quantities Differential length Differential volume A vector whose magnitude is close to zero i.e. dx, dy and dz →0 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 LENGTH, VOLUME AND SURFACE (CYLINDRICAL COORDINATE) All vector components MUST have spatial units i.e. meters, cm, inch etc. DIFFERENTIAL LENGTH, VOLUME AND SURFACE (SPHERICAL COORDINATE) z All vector components MUST have spatial units i.e. meters, cm, inch etc. y x 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 Line integral Definite integral Diagram Maths description Result Information required A measure of the total effect of a given field along a given path Area under the curve 1. Vector field expression A 2. Path expression 1. Function f(x) 2. Integral limits Integral limits depends on path 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 scalars involve inside the integral Result of volume integral is a scalar SURFACE & VOLUME INTEGRAL (CONT.) Surface integral Volume integral Diagram Maths description Result Information required A measure of the total flux from vector field passing through a given surface 1. Vector field expression A 2. Surface expression Integral limits depends on surface A measure of the total effect of a scalar function i.e. temperature, inside a given volume 1. Scalar Function rv 2. Volume expression Integral limits depends on volume PROBLEM 1 Given that F x a x xz a y y a z , calculate the 2 2 circulation of F around the (closed) path shown in the following figure. Solution: ∙ = −1/6 DEL OPERATOR Vector differential operator Must operate on a quantity (i.e. function or vector) to have a meaning Mathematical form: Cartesian Cylindrical Spherical SUMMARY OF GRAD, DIV & CURL Gradient must operate on Scalar f(x,y) Divergence Vector A Curl Vector A Expression (Cartesian) Expression (Cylindrical) Expression (Spherical) Result Vector Scalar Vector SUMMARY OF GRAD, DIV & CURL Gradient Divergence A vector that gives direction of the maximum rate of change of a quantity i.e. temp A scalar that measures the magnitude of a source or sink at a given point Physical meaning ∙<0 Sink i.e. Flux out < flux in ∙<0 Source i.e. Flux out > flux in ∙=0 Incompressible Flux out = flux in Curl A vector operator that describes the rotation/ununiformity of a vector field > 0 RHC rotation < 0 LHC rotation = 0 Irrotational 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 volume integral involving divA 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. PROBLEM 2 (MIDTERM EXAM 2013) Verify the divergence theorem for the vector r2ar within the semisphere. STOKE’S THEOREM 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 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. LAPLACIAN OF A SCALAR FUNCTION U 2 U U 2 x 2 U 2 y 2 U 2 z 2 2 2 2 2, 2, 2 x y z 2 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!! PROBLEM 3 Given that F x y a x y a y , find 2 (a) LF d l Where L is shown in the following figure (b) S F d S Where S is the area bounded by L (c) Is Stokes’s theorem satisfied? 2 1 3