Category Archives: 02 Mathematics

Vector Calculus – Div, grad and curl in suffix notation

Posted on March 17, 2024 by ateixeira

— 9. Grad, Div and Curl revisited in suffix notation — After introducing suffix notation and noting its computing and synthesizing power it is now time to use its power for the differential operators in vector calculus. From now on … Continue reading →

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Vector Calculus – Exercises V

Posted on March 17, 2024 by ateixeira

Exercise 18 Write in suffix notation. As always we will write the equation in suffix notation for the coordinate: Exercise 19 Write in vector notation. Exercise 20 Show that using suffix notation. Exercise 21 Let and be matrices. Show that … Continue reading →

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Vector Calculus – The Kronecker delta and the alternating tensor

Posted on March 16, 2024 by ateixeira

— 7. Kronecker delta — In vector calculus it is important to define the so-called Kronecker delta, which we can think as being identical to the identity matrix. Definition 11 The Kronecker delta is represented by the symbol and by … Continue reading →

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Vector Calculus – Suffix notation

Posted on March 14, 2024 by ateixeira

— 6. Suffix notation — The purpose of this section is to introduce suffix notation and get familiar with it by solving exercises. Suffix notation is a very powerful and agile way of doing calculations when dealing with vectors (and … Continue reading →

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Vector Calculus – Exercises IV

Posted on March 13, 2024 by ateixeira

Exercise 14 Given the scalar field determine and the Laplacian . For the gradient it is Exercise 15 Given the scalar field determine and the Laplacian . And for the Laplacian it is Exercise 16 Find the divergence and the … Continue reading →

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Gradient, Divergence and Curl II

Posted on March 11, 2024 by ateixeira

— 5.4. Divergence — The divergence operation is one of the ways of differentiating a vector field. Let us consider a volume of a rectangular box centered on the point . The lengths of the sides are , and . … Continue reading →

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Vector Calculus – Exercises III

Posted on March 6, 2024 by ateixeira

Exercise 9 Find the gradient of the scalar field and evaluate it at . Hence find the directional derivative of at this point in the direction of the vector . For the gradient it is Hence at point the gradient … Continue reading →

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Gradient, Divergence and Curl I

Posted on March 5, 2024 by ateixeira

— 5. Gradient, Divergence and Curl — — 5.1. Small increments to functions — We saw in previous sections that sometimes we can consider small increments to functions. In this section we will generalize the procedure to functions of more … Continue reading →

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Vector Calculus – Exercises II

Posted on March 5, 2024 by ateixeira

Exercise 5 Evaluate the surface integral of over the surface of defined by with , , with the normal directed in the positive direction. Since the surface is the normal is . Hence Thus the integral is Exercise 6 Find … Continue reading →

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Vector Calculus – Volume Integrals

Posted on March 5, 2024 by ateixeira

— 4.5. Volume integrals — An object of volume with constant mass density has a mass of . If the mass density instead is a function of position, we need to compute the mass of the object in a more … Continue reading →

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