Table of Contents
Table of Contents
"Exercises of Line, Surface and Volume Integrals"
INTRODUCTION
LINE INTEGRALS
SURFACE AND VOLUME INTEGRALS
"Exercises of Line, Surface and Volume Integrals"
"Exercises of Line, Surface and Volume Integrals"
SIMONE MALACRIDA
In this book, exercises are carried out regarding the following mathematical topics:
line integrals of first and second species
integrals of surface area and volume
Initial theoretical hints are also presented to make the performance of the exercises understood.
Simone Malacrida (1977)
Engineer and writer, has worked on research, finance, energy policy and industrial plants.
ANALYTICAL INDEX
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INTRODUCTION
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I – LINE INTEGRALS
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Exercise 17
Exercise 18
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II – SURFACE AND VOLUME INTEGRALS
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
INTRODUCTION
INTRODUCTION
In this workbook some examples of calculations related to line, surface and volume integrals are carried out.
The resolution of these integrals turns out to be the most immediate application of integral calculus applied to differential geometry, especially as regards double and triple integrals.
In order to understand in more detail what is explained in the resolution of the exercises, the theoretical context of reference is recalled at the beginning of each chapter.
What is presented in this workbook is generally addressed in advanced mathematical analysis courses (analysis 2).
It goes without saying that, to fully understand the above, it is necessary to already have a knowledge of double and triple integrals and of the parametrization of curves.
I
LINE INTEGRALS
LINE INTEGRALS
We define line integral of the first kind as an integral of a scalar field along a curve defined in parametric form in an open set.
The line integral of the first kind is:
Where ds is the curvilinear abscissa, 
it is the curve along which the integral is calculated and it 
represents the parametrization of the scalar function.
The line integral of the first kind enjoys the properties of linearity,additivity and monotonicity.
In addition, the following surcharges apply:
If the domain of the function is R, the curvilinear integral of the first kind is the normal Riemann integral.
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The line integral of the second kind is the integral of a vector field along a curve. This integral is equal to the scalar product between the vector field and the unit vector tangent to the curve:
The integral is also called work because in physics it expresses the work of a force along a path. This integral has the same properties as that of the first kind, moreover it is independent of the parametric representation adopted, except for the direction of travel which causes the sign to change.
Exercise 1
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After verifying that the support of the curves is contained in the domain of the functions, compute the following line integral of the first kind:
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The function is defined in:
The support of the curve is given by:
And it is contained in the domain.
Being:
The curve is differentiable with a continuous derivative and therefore is regular.
Furthermore it is noted that:
Substituting, we have:
Exercise 2
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After verifying that the support of the curves is contained in the domain of the functions, compute the following line integral of the first kind:
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The function is defined in:
The support of the curve is given by the arc of the circumference with center O(0,0) and radius 1 present in the I and in the IV quadrant, in fact:
And it is contained in the domain.
Being:
The curve is differentiable with a continuous derivative and therefore is regular.
Furthermore it is noted that:
Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 21.04.2023
ISBN: 978-3-7554-3976-9
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