Vector_Analysis_2 - THIS MATERIAL WILL HELP YOU TO PREPARE...

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Unformatted text preview: THIS MATERIAL WILL HELP YOU TO PREPARE FOR THE BASIC CONCEPTS IN MATHEMATICS FOR ENGINEERING ELECTROMAGNETICS SOME BASIC MATHEMATICAL CONCEPTS FOR ENGINEERING ELECTROMAGNETICS 1 KEY TOPICS: x DEL OPERATOR x GRADIENT x DIVERGENCE x CURL x LAPLACIAN SOME BASIC MATHEMATICAL CONCEPTS FOR ENGINEERING ELECTROMAGNETICS 2 DEL OPERATOR 5 (del) operator provides a means of describing the SPATIAL DERIVATIVES of a SCALAR or a VECTOR field. Del operator is useful in defining the following operations: 1. The GRADIENT of a SCALAR field 2. The DIVERGENCE of a VECTOR field 3. The CURL of a VECTOR field 4. The LAPLACIAN of a SCALAR field SOME BASIC MATHEMATICAL CONCEPTS FOR ENGINEERING ELECTROMAGNETICS 3 GRADIENT Gradient of a SCALAR field V , written as 5 V , is a vector that represents both the MAGNITUDE and the DIRECTION of the maximum space rate of increase of V . Gradient in Cartesian coordinates is defined as: 5 V x, y, z ` a = a x jjjjjjjjj k V x ffffffffff + a y jjjjjjjjjj k V y ffffffffff + a z jjjjjjjjjj k V z ffffffffff Gradient in Cylindrical coordinates is defined as: 5 V r, , z b c = a r jjjjjjjjj k V r ffffffffff + a jjjjjjjjjj k 1 r fff V ffffffffff + a z jjjjjjjjjj k V z ffffffffff Fundamental properties of Gradient of Scalar field V : 1. Magnitude of 5 V equals the maximum rate of change in V per unit distance 2. 5 V points in the direction of the maximum rate of increase in V 3. 5 V at any point is perpendicular to the constant V surface that passes through that point SOME BASIC MATHEMATICAL CONCEPTS FOR ENGINEERING ELECTROMAGNETICS 4 Example: Q.1. Find the gradient of the following scalar fields: a. V = x 2 y + xyz b. U = r 2 z cos 2 Solution: a. b. 5 V x, y, z ` a = a x jjjjjjjjj k V x ffffffffff + a y jjjjjjjjjj k V y ffffffffff + a z jjjjjjjjjj k V z ffffffffff 5 V x, y, z ` a = 2 xy + yz b c a x jjjj jjjj j k + x 2 + xz b c a y jjjj jjjj jj k + xy a z jjjj jjjj jj k # 5 V x, y, z ` a = y 2 x + z ` a a x jjjjjjjjj k + x x + z ` a a y jjjjjjjjjj k + xy a z jjjjjjjjjj k 5 U r, , z b c = a r jjjjjjjjj k U r fffffffffff + a jjjjjjjjjj k 1 r fff U fffffffffff + a z jjjjjjjjjj k U z fffffffffff # 5 U r, , z b c = 2 r z cos 2 a r jjjjjjjjj k @ 2 r z sin 2 a jjjjjjjjjj k + r 2 cos 2 a z jjjjjjjjjj k SOME BASIC MATHEMATICAL CONCEPTS FOR ENGINEERING ELECTROMAGNETICS 5 DIVERGENCE Divergence of VECTOR A jjjjjjj k , written as 5A A jjjjjjj k , at point P is a measure of the net outward flow of flux per unit volume as the volume shrinks about P Examples : air from balloon; car exhaust; kitchen sink; etc. (a) Positive Divergence (b) Negative Divergence (c) Zero Divergence (a) Divergence > 0 because the vector diverges (spreads out) at P or more flux is flowing out than flowing in (in this case P is known as SOURCE POINT ) (b) Divergence < 0 because the vector converges at P or more flux is flowing in than flowing out (in this case P is known as...
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Vector_Analysis_2 - THIS MATERIAL WILL HELP YOU TO PREPARE...

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