hence the Laplacian of a scalar is a scalar the Laplacian of a vector is a

Hence the laplacian of a scalar is a scalar the

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; hence the Laplacian of a scalar is a scalar, the Laplacian of a vector is a vector, the Laplacian of a rank-2 tensor is a rank-2 tensor, and so on. F. Divergence Theorem The divergence theorem, which is also known as Gauss theorem , is a mathematical statement of the intuitive idea that the integral of the divergence of a vector field over a given volume is equal to the total flux of the vector field out of the surface enclosing the volume. Symbolically, the divergence theorem states that: ˚ V ∇ · A = ¨ S A · n (41) [15] This operator is also known as the harmonic operator.
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1.3.2 Vector Algebra and Calculus 33 where A is a differentiable vector field, V is a bounded volume in an n D space enclosed by a surface S , and are volume and surface elements respectively, and n is a variable unit vector normal to the surface. The divergence theorem is useful for converting volume integrals into surface in- tegrals and vice versa. In many cases, this can result in a considerable simplification of the required mathematical work when one of these integrals is easier to manipulate and evaluate than the other, or even overcoming a mathematical hurdle when one of the inte- grals cannot be evaluated analytically. Moreover, the divergence theorem plays a crucial role in many mathematical proofs and theoretical arguments in mathematical and physical theories. G. Stokes Theorem Stokes theorem is a mathematical statement that the integral of the curl of a vector field over an open surface is equal to the line integral of the field around the perimeter surrounding the surface, that is: ¨ S ( ∇ × A ) · n = ˆ C A · d r (42) where A is a differentiable vector field, C symbolizes the perimeter of the surface S , d r is a vector element tangent to the perimeter, and the other symbols are as defined in the divergence theorem. The perimeter should be traversed in a sense related to the direction of the normal vector n by the right hand twist rule , that is when the fingers of the right hand twist in the sense of traversing the perimeter the thumb will point approximately in the direction of n , as seen in Figure 10. Similar to the divergence theorem, Stokes theorem is useful for converting surface integrals into line integrals and vice versa, which is useful in many cases for reducing the amount of mathematical work or overcoming technical and mathematical difficulties.
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1.3.3 Matrix Algebra 34 x 1 x 2 x 3 n C S d r Figure 10: Illustration of Stokes integral theorem (left frame) with the right hand twist rule (right frame). Stokes theorem is also crucial in the development of many proofs and theoretical arguments in mathematics and science. 1.3.3 Matrix Algebra There is a close relation between rank-2 tensors and square matrices where the latter usually represent the former. Hence, there are many ideas, techniques and notations which are common or similar between the two subjects. We therefore provide in this subsection a set of short introductory notes about matrix algebra to supply the reader
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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