Table of Contents
Table of Contents
“The Book of Mathematics: Volume 2”
MULTIVARIABLE REAL FUNCTIONS
DIFFERENTIAL GEOMETRY
MULTIVARIABLE INTEGRAL CALCULATION
INTEGRALS OF SURFACE AND VOLUME
TENSORS AND TENSORIAL MATHEMATICS
COMPLEX ANALYSIS
FUNCTIONAL ANALYSIS
TRANSFORM
DISTRIBUTIONS
ORDINARY DIFFERENTIAL EQUATIONS
PARTIAL DIFFERENTIAL EQUATIONS
“The Book of Mathematics: Volume 2”
“The Book of Mathematics: Volume 2”
SIMONE MALACRIDA
Most of mathematics is presented in this book, starting from the basic and elementary concepts to probing the more complex and advanced areas.
Mathematics is approached both from a theoretical point of view, expounding theorems and definitions of each particular type, and on a practical level, going on to solve more than 1,000 exercises.
The approach to mathematics is given by progressive knowledge, exposing the various chapters in a logical order so that the reader can build a continuous path in the study of that science.
The entire book is divided into three distinct sections: elementary mathematics, the advanced mathematics given by analysis and geometry, and finally the part concerning statistics, algebra and logic.
The writing stands as an all-inclusive work concerning mathematics, leaving out no aspect of the many facets it can take on.
Simone Malacrida (1977)
Engineer and writer, has worked on research, finance, energy policy and industrial plants.
ANALYTICAL INDEX
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26 – MULTIVARIABLE REAL FUNCTIONS
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27 – DIFFERENTIAL GEOMETRY
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28 – MULTIVARIABLE INTEGRAL CALCULATION
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29 – INTEGRALS OF SURFACE AND VOLUME
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30 – TENSORS AND TENSORIAL MATHEMATICS
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31 – COMPLEX ANALYSIS
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32 – FUNCTIONAL ANALYSIS
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33 - TRANSFORM
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34 - DISTRIBUTIONS
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35 – ORDINARY DIFFERENTIAL EQUATIONS
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36 – PARTIAL DIFFERENTIAL EQUATIONS
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26
MULTIVARIABLE REAL FUNCTIONS
MULTIVARIABLE REAL FUNCTIONS
Introduction
Functions of real variables with several variables are an extension of what has been said for real functions with one variable.
Almost all the properties mentioned for one-variable functions remain valid (such as injectivity, surjectivity and bijectivity), except the ordering property which is not definable.
The domain of a multivariate function is given by the Cartesian product of the domains calculated on the single variables.
A level set, or level curve, is the set of points such that:
The level set with c=0 is used to analyze the sign of the function in the domain.
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Operations
The topological definition of limit is the same as that given for one-variable functions, the metric definition changes as follows:
The limit exists if its value does not depend on the direction in which it is calculated.
The same applies to continuity.
A function is said to be continuous separately with respect to one of its variables if it is continuous as a function of the single variable, keeping the other constants.
Separate continuity is a weaker condition than global continuity across all variables.
For a function of several variables, however, there are different concepts of derivative.
We call partial derivative the derivative carried out only on one of the variables, always defining the derivative as the limit of an incremental ratio.
To distinguish the partial derivative from the total one, the symbol is used 
.
Partial derivatives of higher order return the order to the exponent of that symbol.
A point is said to be simple if the first partial derivatives are continuous and not zero, but if one of the derivatives is zero or does not exist, the point is said to be singular.
Partial differentiability implies separate continuity.
By extending the concept of partial derivative from a path along the coordinate axes to any path, we have the directional derivative.
Once a generic unit vector is defined, the directional derivative along that vector is given by:
The directional derivative indicates the rate of change of the function with respect to the given direction.
The derivative of a function with several variables which takes into account the mutual dependence of the variables themselves is defined as the total derivative.
For example we have:
However, differentiability is not a sufficient condition for continuity.
A sufficient condition is instead given by differentiability.
A function of several variables is differentiable at a point of in an open set of n-dimensional Euclidean space R if there exists a linear map L such that the following relation holds:
The total prime differential is given by the following product:
While the total derivative is given by 
.
The function is differentiable if it is differentiable at every point in its domain.
The total differential theorem states that a function is differentiable at a point if all partial derivatives exist in a neighborhood of the point and if these partial derivatives are continuous.
If the application is also continuous, the function is said to be continuously differentiable.
The total prime differential can also be expressed as:
Higher-order total differentials can be expressed as follows, for a function of two variables:
We call mixed derivatives the derivatives of order higher than the first which foresee the derivation of variables different from each other.
For a function of two variables defined on an open set, if it admits continuous mixed second derivatives, Schwarz's theorem holds according to which the order of the derivation can be inverted without changing the result:
If a function is differentiable at a point, then all partial derivatives computed at that point exist and are continuous.
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Jacobian, Hessian and Nabla algebra
The linear map defined as the sum of the first partial derivatives is a matrix m rows n columns called the Jacobian matrix and is exactly the equivalent of the previously mentioned linear map L:
If m=1, the Jacobian matrix reduces to an n-dimensional vector called gradient which indicates the direction of maximum slope of the graph of the function at a point.
If n=1 the function parametrizes a curve and its differential is a function that indicates the direction of the tangent line to the curve at the point.
If m=n=1 the condition of differentiability coincides with that of differentiability and the Jacobian matrix is reduced to a number, equal to the derivative of the function at that point.
If m=n the Jacobian matrix is square and its determinant is known as the Jacobian.
The inverse function theorem states that a continuously differentiable function is invertible if and only if its Jacobian determinant is nonzero.
If a function of several variables is differentiable, then the directional derivative exists and is equal to the scalar product between the gradient with respect to the single variable and the versor itself.
The directional derivative therefore takes on a maximum value when the gradient and the unit vector are parallel and in agreement, a minimum value when they are parallel and discordant, and a null value when they are perpendicular.
A differential is said to be exact if and only if it is integrable, ie if it can be expressed as a function of the second class of simply connected continuity (in other words, Schwarz's theorem must hold).
We define gradient as the quantity which, multiplied according to the scalar product with any vector, returns the directional derivative of the function with respect to the vector.
The gradient is a vector field and, in the case of a Cartesian reference system, it is the sum of the products between the first partial derivatives and versors:
Where in the second member there is the notation according to the nabla operator.
This differential operator is defined as follows:
We define the divergence of a continuous and differentiable vector field as the scalar function given by the dot product between the operator nabla and the vector field:
We define curl of a continuous and differentiable vector field, a vector field given by the vector product between the operator nabla and the field itself:
We define Laplacian the square of the nabla operator equal to:
Some properties of the nabla operator are as follows:
If all second partial derivatives exist, we define the Jacobian matrix of the gradient as Hessian of the function:
If all second derivatives are continuous, Schwarz's theorem holds and the Hessian matrix is symmetric.
If the gradient of the function is zero at a point then that point is called a critical point.
If at that point also the determinant of the Hessian matrix is zero then the critical point is called degenerate.
For a non-degenerate critical point, if the Hessian matrix is positive definite then the function has a local minimum at that point, if instead it is negative definite there is a local maximum.
If the Hessian matrix has all non-zero eigenvalues, and they assume both positive and negative values, that point is called a saddle point.
In all other cases, for example for positive or negative semidefinite Hessian matrices, nothing can be said about the presence of stationary points.
Search for stationary points and method of Lagrange multipliers
A necessary condition for the search for constrained maxima and minima is the so-called Lagrange multiplier method.
For a two-dimensional function, this method states that the necessary condition for having a constrained extremum is that:
The values of 
are precisely the Lagrange multipliers since the function h can be defined as the Lagrangian of the system.
A practical case of application of this formalism is that of Lagrangian mechanics in which the equations of motion are obtained by finding the stationary points of an integral, called action.
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Implicit functions
Implicit functions are functions of the type:
For two-dimensional functions the following Dini theorem holds.
Considering a continuously differentiable function defined on an open set and a non-empty set in which the function f(x,y) is zero, then there exists a point in this set where the following relation holds:
If this point is not critical, i.e. the inequality holds:
Then there exists a neighborhood of this point such that the set given by the intersection of this neighborhood and the set in which the non-critical point is located represents the graph of a differentiable function.
This is equivalent to saying that there is a single explicit function of the type y=y(x) or x=x(y) which relates the two unknowns.
This theorem therefore provides a sufficient condition for the explicitation of the implicit functions.
In multiple dimensions, the function variables can be divided into two blocks, one up to the nth degree and one up to the mth degree, as follows:
The Jacobian matrix computed in the n+m-dimensional open set can be divided into two blocks, recalling the division of variables:
Assuming that X is invertible.
The implicit function theorem states that there is a unique explicitation of the function f(x,y)=0. This function g(y)=x is continuously differentiable and the relation holds:
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Exercises
Exercise 1
Determine the domain and partial derivatives of the following function:
The domain is given by the non-zero denominator, so:
The partial derivatives are simply:
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Exercise 2
Determine the domain and partial derivatives of the following function:
The domain is given by the denominator other than zero and the argument of the tangent different from 90° and its multiples, therefore:
The partial derivatives are simply:
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Exercise 3
Determine the domain and partial derivatives of the following function:
The domain is given by the non-zero denominator and the argument of the logarithm greater than zero, thus:
The partial derivatives are simply:
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Exercise 4
Determine the domain and partial derivatives of the following function:
The domain is given by the non-zero denominator and the root greater than or equal to zero, thus:
The partial derivatives are simply:
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Exercise 5
Determine the local and absolute minimum and maximum points for the following function of two variables on the specified set:
The function is of class 
.
The given set is compact.
By Weierstrass's theorem there exist a maximum and a minimum of the function in the set.
We use the Lagrange multiplier method to find these points.
Place:
We look for the stationary points of the function:
This means that:
Or:
Expanding the accounts we have:
The stationary points of this function are therefore:
Returning to the starting function, the stationary points are:
Since you have:
The first point is the absolute maximum, while the second is the absolute minimum.
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Exercise 6
Determine the local and absolute minimum and maximum points for the following function of two variables on the specified set:
The function is of class 
.
The given set is compact.
By Weierstrass's theorem there exist a maximum and a minimum of the function in the set.
We have:
Place:
It is seen that:
This means that:
So the first two points are the absolute minimum, while the second two are the absolute maximum.
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Exercise 7
Determine the local and absolute minimum and maximum points for the following function of two variables on the specified set:
The function is of class 
.
The given set is compact.
By Weierstrass's theorem there exist a maximum and a minimum of the function in the set.
We use the Lagrange multiplier method to find these points.
Place:
We look for the stationary points of the function:
This means that:
Or:
Expanding the accounts we have:
The stationary points of this function are therefore:
Returning to the starting function, the stationary points are:
Since you have:
The first point is the absolute minimum, while the second is the absolute maximum.
Exercise 8
Determine the local and absolute minimum and maximum points for the following function of two variables on the specified set:
The function is of class .
The given set is compact (check that it is closed and bounded using the topological properties of complementarity).
By Weierstrass's theorem there exist a maximum and a minimum of the function in the set.
We use the Lagrange multiplier method to find these points.
We look for the stationary points of the function:
This means that:
Or:
Expanding the accounts we have:
The stationary points of this function are therefore:
Returning to the starting function, the stationary points are:
Since you have:
The first two points are the all time low, while the second two is the all time high.
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Exercise 9
Determine the local and absolute minimum and maximum points for the following function of two variables on the specified set:
The function is of class 
.
The given set is compact.
By Weierstrass's theorem there exist a maximum and a minimum of the function in the set.
We initially look for the inner extremum points of M:
The extreme points are to be found among the stationary ones, i.e.:
So we have that:
The only internal stationary point is therefore:
To establish whether it is maximum or minimum or saddle, we calculate the Hessian matrix:
Then the Hessian matrix at the point is given by:
So the point is of local minimum and the function is valid, at this point:
We now look for the stationary points on the edge:
We have that:
Place:
The extreme points are to be found between the stationary points and the extremes of the interval [-1,1] where this new function is defined.
Since:
It follows that a stationary point is:
Studying the sign of the derivative we see that this point is a minimum. It is a local minimum. At the ends of the interval we have:
So x=-1 is an absolute maximum point, while x=1 is a local maximum point. Comparing the values of the minima, it is found that:
It is an absolute minimum point.
Exercise 10
Determine the local and absolute minimum and maximum points for the following function of two variables on the specified set:
The function is of class .
The given set is compact.
By Weierstrass's theorem there exist a maximum and a minimum of the function in the set.
We initially look for the inner extremum points of M:
The extreme points are to be found among the stationary ones, i.e.:
So we have that:
The only internal stationary point is therefore:
To establish whether it is maximum or minimum or saddle, we calculate the Hessian matrix:
Then the Hessian matrix at the point is given by:
So the point is of local minimum and the function is valid, at this point:
We now look for the stationary points on the edge:
We have that:
Place:
The extreme points are to be found between the stationary points and the extremes of the interval [-2,2] where this new function is defined. Since:
It follows that there are no stationary points.
Furthermore, since the derivative is always negative in the given interval, it follows that x=2 is a point of absolute maximum and ex=-2 of absolute minimum for this function.
So (-2,0) is an absolute minimum point for f on the edge, while (2,0) is an absolute maximum point for f on the edge. Being:
We see that (1,0) is the point of absolute minimum for f over the whole set M.
Exercise 11
Determine the local and absolute minimum and maximum points for the following function of three variables on the specified set:
The function is of class 
.
The given set is compact.
By Weierstrass's theorem there exist a maximum and a minimum of the function in the set.
We initially look for the inner extremum points of M:
The extreme points are to be found among the stationary ones, i.e.:
So we have that:
Therefore:
The internal stationary points are:
These points are of absolute minimum, as:
We observe that also the points
They are stationary and of absolute minimum, but they are not internal to M.
We now look for the stationary points on the edge:
We proceed with the method of Lagrange multipliers.
Place:
We have that:
The stationary points are equivalent to solving:
Doing the calculations, we have:
The stationary points of this function are:
The stationary points of f are therefore:
The former are the points of absolute maximum, while we find the latter as those of absolute minimum.
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Exercise 12
Determine the local and absolute minimum and maximum points for the following function of three variables on the specified set:
The function is classy
.
The set M is closed and unbounded, so Weierstrass's theorem cannot be applied.
Nevertheless, using the definition of limit, we can write that:
The function f is symmetric in M with respect to the xy plane, i.e.:
In other words, if we prove that there is a maximum, there must also be a minimum.
Taking any R value, it can be written that:
The intersection between M and such a set:
It is a compact and non-empty set, and also contains the point in question.
In this set, Weierstrass's theorem can be applied for which the function admits a maximum and a minimum.
We initially look for the inner extremum points of M:
The extreme points are to be found among the stationary ones, i.e.:
So we have that:
Therefore:
The internal stationary points are:
We observe that also the points
They are stationary, but they are not internal to M.
All these points are neither minimum nor maximum, in fact:
But it is also valid:
We now look for the stationary points on the edge:
We proceed with the method of Lagrange multipliers.
Place:
We have that:
The stationary points are equivalent to solving:
Doing the calculations, we have:
The stationary points of this function are:
The stationary points of f are therefore:
We have that:
Therefore, the absolute maxima are:
While the absolute minimums:
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Exercise 13
Determine the local and absolute minimum and maximum points for the following function of three variables on the specified set:
The function is classy
The set M is compact.
We can apply the Weierstrass theorem that the function admits a maximum and a minimum.
We initially look for the inner extremum points of M:
The extreme points are to be found among the stationary ones, i.e.:
So we have that:
The function does not admit stationary points, therefore not even maxima and minima.
We now look for the stationary points on the edge:
We proceed with the method of Lagrange multipliers.
Place:
We have that:
The stationary points are equivalent to solving:
Doing the calculations, we have:
The stationary points of this function are:
The stationary points of f are therefore:
We have that:
Therefore, the absolute maxima are:
While the absolute minimums:
As for the points:
We note that, based on the choice of neighborhoods, both of the following relations hold:
And therefore these points are neither maximum nor minimum.
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Exercise 14
Determine the local and absolute minimum and maximum points for the following function of three variables on the specified set:
The function is classy
The set M is closed and bounded, therefore compact.
We can apply the Weierstrass theorem that the function admits a maximum and a minimum.
We proceed with the method of Lagrange multipliers.
Place:
We have that:
The stationary points are equivalent to solving:
Doing the calculations, we have:
The stationary points of this function are:
The stationary points of f are therefore:
We have that:
Therefore, the absolute maxima are:
While the absolute minimums:
As for the points:
We note that, based on the choice of neighborhoods, both of the following relations hold:
And therefore these points are neither maximum nor minimum.
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Exercise 15
Determine the local and absolute minimum and maximum points for the following function of three variables on the specified set:
The function is classy
The set M is closed and unbounded, so Weierstrass's theorem cannot be applied.
Nevertheless, using the definition of limit, we can write that:
The function f is symmetric in M with respect to the planes xy=1 and x+y=1, i.e.:
In other words, if we prove that there is a maximum, there must also be a minimum.
Taking any R value, it can be written that:
The intersection between M and such a set:
It is a compact and non-empty set, and also contains the point in question.
In this set, Weierstrass's theorem can be applied for which the function admits a maximum and a minimum.
We initially look for the inner extremum points of M:
The extreme points are to be found among the stationary ones, i.e.:
So we have that:
Therefore:
The internal stationary points are:
All these points are neither minimum nor maximum, in fact:
But it is also valid:
We now look for the stationary points on the edge:
We proceed with the method of Lagrange multipliers.
Place:
We have that:
The stationary points are equivalent to solving:
Doing the calculations, we have:
The stationary points of this function are:
The stationary points of f are therefore:
We have that:
Therefore, the absolute maxima are:
While the absolute minimums:
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Exercise 16
Determine the local and absolute minimum and maximum points for the following function of three variables on the specified set:
The function is classy
The set M is closed and unbounded, so Weierstrass's theorem cannot be applied.
Nevertheless, using the definition of limit, we can write that:
The function f is symmetric in M with respect to the planes xy=0 and x+y=0, i.e.:
In other words, if we prove that there is a maximum, there must also be a minimum.
Taking any R value, it can be written that:
The intersection between M and such a set:
It is a compact and non-empty set, and also contains the point in question.
In this set, Weierstrass's theorem can be applied for which the function admits a maximum and a minimum.
We initially look for the inner extremum points of M:
The extreme points are to be found among the stationary ones, i.e.:
So we have that:
The internal stationary points are:
All these points are neither minimum nor maximum, in fact:
But it is also valid:
We now look for the stationary points on the edge:
We proceed with the method of Lagrange multipliers. Place:
We have that:
The stationary points are equivalent to solving:
Doing the calculations, we have:
The stationary points of this function are:
The stationary points of f are therefore:
We have that:
Therefore, the absolute maxima are:
While the absolute minimums:
27
DIFFERENTIAL GEOMETRY
DIFFERENTIAL GEOMETRY
Introduction
Differential geometry is concerned with the study of geometric objects through mathematical analysis.
At the basis of differential geometry is the notion of differentiable manifold which generalizes both the concepts of curve and surface in a space of any dimension and the approach given by topological manifolds.
The differentiable manifolds also represent the connection with the differential topology in fact they are topological spaces and, locally, Euclidean spaces which are connected to each other through differentiable functions.
Considering a topological variety, the open sets that make up its cover can be related to an open set of Euclidean space through a set of homeomorphisms to which we give the name of atlas (while the single homeomorphism is called map).
The composition of functions consisting of a card and its inverse function is called a transition function.
A topological manifold is differentiable if the transition function is differentiable.
A differentiable submanifold in a differentiable manifold is a subset which is described as zero of a differentiable function.
In the case of submanifolds with codomain equal to the set of real numbers we speak of a hypersurface and the condition of differentiability is equivalent to requiring that the gradient of the submanifold on each map is everywhere different from zero.
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Operations
We define exterior product in a vector space, a product of associative and bilinear vectors:
are linearly dependent
A differential form defined on an open set is given by the following expression:
With functions given by differentiable functions.
The form
Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 18.04.2023
ISBN: 978-3-7554-3917-2
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