Understanding the Del Operator in Vector Calculus
Explore the Del operator and its applications in vector calculus, including concepts like gradients, divergence, and curl. Learn how to express these operations in different coordinate systems and understand the fundamental properties of the gradient. Dive into the mathematical expressions and theorems related to the Del operator.
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Presentation Transcript
Del operator Part I
Del operator: The del operator, written ?, is the vector differential operator. In Cartesian coordinates, ? ????+ ? ????+? ? = ???? The operator is useful in defining 1. The gradient of a scalar V, written, as ?V 2. The divergence of a vector A, written as ? A 3. The curl of a vector A, written as ? X A 4. The Laplacian of a scalar V, written as ??V 2
In cylindrical coordinates as ? ????+1 ? ? +? ? ? = ???? ? In spherical coordinates: ? ????+1 ? 1 ? ? = ????+ ? ? ? ?sin? 3
Gradient of a scalar: The gradient of a scalar field V is a vector that represents both the magnitude and the direction of the maximum space rate of increase of V. FIGURE Gradient of a scalar. (V1, V2, and V3 are contours on which V is constant.) 4
Mathematical expression for the gradient can be obtained by evaluating the difference in the field dV between points P1and P2of Figure ?? =?? ???? +?? ???? +?? ???? ?? ????+?? ????+?? = ????. ????+ ????+ ???? For convenience, let ?? ????+?? ????+?? ? = ???? Then ?? = ?.?? = ? cos??? ?? ??= ? cos? where ??is the differential displacement from P1 to P2and ?is the angle between G and ??. ?? ?? ??? =?? ??= ? where dV/dn is the normal derivative. 5
Thus G has its magnitude and direction as those of the maximum rate of change of V. By definition, G is the gradient of V. Therefore ???? ? = ?? =?? ????+?? ????+?? ???? For Cartesian coordinates ?? =?? ????+?? ????+?? ???? For Cylindrical coordinates, ?? =?? ????+1 ?? ? ? +?? ???? ? For Spherical coordinates, ?? =?? ????+1 ?? ????+ 1 ?? ? ? ? ?sin? 6
Fundamental properties of the gradient: The magnitude of ?? equals the maximum rate of change in V per unit distance. ?? points in the direction of the maximum rate of change in V. ?? at any point is perpendicular to the constant V surface that passes through that Point. The projection (or component) of ??in the direction of a unit vector a is ?? a and is called the directional derivative of V along a. This is the rate of change of V in the direction of a. If A = ??, V is said to be the scalar potential of A. 7
Divergence of a vector and divergence theorem: The divergence of A at a given point P is the outward flux per unit volume as the volume shrinks about P. ??.?? ? ??? ? = ?.? = lim ? ? FIGURE Illustration of the divergence of a vector field at P: (a) positive divergence, (b) negative divergence, (c) zero divergence. 8
For Cartesian coordinates ?? =??? ??+??? ??+??? ?? For Cylindrical coordinates, ?? =1 ? ??(???) +1 ?? ? +??? ? ? ?? For Spherical coordinates, ?? =1 ? ??(?2??) + 1 ? 1 ? ??(??sin?) + ? ? ?2 ?sin? ?sin? 9
Properties of the divergence of a vector field: It produces a scalar field (because scalar product is involved). The divergence of a scalar V, div V, makes no sense. ? (A + B) = ? A + ? B ? (VA) = V? A + A ?V From the definition of the divergence of A ??? = ???? ? ? This is called the divergence theorem, otherwise known as the Gauss- Ostrogradsky theorem. The divergence theorem states that the total outward flux of a vector field A through the closed surface S is the same as the volume integral of the divergence of A. 10
References 1. M.N. Sadiku, Elements of Electromagnetics, New York: Oxford University Press, 2000.
