149 If a vector field is given in the Cartesian coordinates by A 2 y 3 x 1 5 z

# 149 if a vector field is given in the cartesian

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1.49 If a vector field is given in the Cartesian coordinates by A = (2 y, - 3 x, 1 . 5 z ) , verify Stokes theorem for a hemispherical surface x 2 + y 2 + z 2 = 9 for z 0 . 1.50 Make a simple sketch to demonstrate Stokes theorem with sufficient explanations and definitions of the symbols involved. 1.51 Give concise definitions for the following terms related to matrices: matrix, square matrix, main diagonal, trailing diagonal, transpose, identity matrix, unit matrix,
1.4 Exercises 45 singular, trace, determinant, cofactor, and inverse. 1.52 Explain the way by which matrices are indexed. 1.53 How many indices are needed in indexing a 2 × 3 matrix, an n × n matrix, and an m × k matrix? Explain, in each case, why. 1.54 Does the order of the matrix indices matter? If so, what is the meaning of changing this order? 1.55 Is it possible to write a vector as a matrix? If so, what is the condition that should be imposed on the indices and how many forms a vector can have when it is written as a matrix? 1.56 Write down the following matrices in a standard rectangular array form (similar to the examples in Eq. 43) using conventional symbols for their entries with a proper indexing: 3 × 4 matrix A , 1 × 5 matrix B , 2 × 2 matrix C , and 3 × 1 matrix D . 1.57 Give detailed mathematical definitions of the determinant, trace and inverse of matrix, explaining any symbol or technical term involved in these definitions. 1.58 Find the following matrix multiplications: AB , BC , and CB where: A = 9 . 6 6 . 3 - 22 - 3 . 8 2 . 5 2 . 9 - 6 3 . 2 7 . 5 B = - 3 . 8 - 2 . 0 4 . 6 11 . 6 12 . 0 25 . 9 C = 3 8 . 4 61 . 3 - 5 - 33 5 . 9 1.59 Referring to the matrices A , B and C in the previous exercise, find all the permuta- tions (repetitive and non-repetitive) involving two of these three matrices, and classify them into two groups: those which do represent possible matrix multiplication and those which do not.
1.4 Exercises 46 1.60 Is matrix multiplication associative? commutative? distributive over matrix addition? 1.61 Calculate the trace, the determinant, and the inverse (if the inverse exists) of the following matrices: D = 3 . 2 2 . 6 1 . 6 12 . 9 - 1 . 9 2 . 4 - 11 . 9 33 . 2 - 22 . 5 E = 5 . 2 2 . 7 3 . 6 - 10 . 4 - 5 . 4 - 7 . 2 - 31 . 9 13 . 2 - 23 . 7 1.62 Which, if any, of the matrices D and E in the previous exercise is singular? 1.63 Select from the Index six terms connected to the special matrices which are defined in § Special Matrices.
Chapter 2 Tensors In this chapter, we present the essential terms and definitions related to tensors, the conventions and notations which are used in their representation, the general rules that govern their manipulation, and their main types and classifications. We also provide some illuminating examples of tensors of various complexity as well as an overview of their use in mathematics, science and engineering.

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• Summer '20
• Rajendra Paramanik
• Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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