EE2011 Lecture 1 - Review of Vector Calculus

In general f and will not be constant but changing

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Unformatted text preview: 5b The previous example is a very simple one. In general, F and will not be constant but changing with position (x, y, z). This will require us to do the real integration for each segment. More exercises will be given in the tutorial questions. 15c Vector Surface Integrals Definition of a vector area: S = vector area ˆ =S a a ˆ Direction of this unit ˆ normal vector = a Surface area = S 15d F ds S FA ds SA FB ds SB FC ds SC FA S A FB S B FC SC ˆ S AFA a S B FB b SC FC c ˆ ˆ S vector area S A S B SC ˆ S Aa S B b SC c ˆ ˆ z c ˆ FC SC a ˆ x 15e SB ˆ b SA F B F A y Example FA FB S 3y ˆ SB 2y m ˆ FC F ds 2x ˆ SA z ˆ Sc 5z m ˆ FB ds FC ds FA ds SA SB 4x m 2 ˆ 2x 4x 3y 2y 1z 5z ˆˆ ˆˆˆˆ 8 6 5 19 15f 2 z SC x 2 z ˆ FC S C x ˆ SB y ˆ SA F B F A y The previous example is a very simple one. In general, F and S will not be constant but changing with position (x, y, z). This will require us to do the real integration for each surface area. More exercises will be given in the tutorial questions. 15g Gradient of a Scalar Field • Basic Concept of the Gradient – Let us assume a temperature T as a function of space parameters – Gradient is an extension of the derivative dT/dz – However, if the T is a function of 3-D position (x, y, z), then we need to use Gradient to describe the increase if the temperature T as a function of (x, y, z) Prof Joshua Le-Wei Li, EM Research Group Prof 16 EE2011: Engineering Electromagnetics Electromagnetics Gradient of a Scalar Field • Basic Concept of the Gradient – Del or gradient operator dl dT ˆ ˆ ˆ xdx ydy zdz T T dy dx y x T dz z T ˆ x x T T T ˆ ˆ ˆ dl dl z dl y x z y x T ˆ...
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