The document is a report submitted by three students on surface and volume integrals and linear systems in real world problems. It discusses line, surface, and volume integrals and defines linear systems. It provides examples of applying integrals in electrostatics, magnetism, and gravity. It also discusses applications of linear systems and integrals in areas like fluid dynamics, mass continuity, and conservation of momentum. The conclusion reiterates that linear systems can have zero, one, or infinite solutions depending on consistency and independence, and methods for solving systems like graphing, substitution, and addition/subtraction are introduced.

1. Career Point Cares
A REPORT
ON
“SURFACE AND VOLUME INTEGRALS AND LINEAR SYSTEM IN
REAL WORLD PROBLEMS”
SUBMITTED TO:- Dr. SONA RAJ SUBMITTED BY:-
JABI KHAN (K12251)
RAVI SHANKER BORDIYA (K12256)
HARSH SHARMA (K12401)
COURSE:- B.tech (civil)

2. Career Point Cares
CONTENTS
Introduction
 Vector integral
• Line integral
• Surface integral
• Volume integral
 Linear system
• Realworld application
• Applications of branch
• Conclusion
• Referance

3. Career Point Cares
introduction
• Line integral:- the integral, taken along a line, of any function that has a
continously varying value along that line.
• Surface integral:- in mathematucs , a surface integral is a generalization
of multiple integral to integration over surfaces.it can be thought of as the double
integral analog of the line integral.
• Volume integral:-in particular, in multivariable calculus- a volumeb
integral refers to an integral over a 3-diamensional domain, that is it is a special case
of multiple integrals. Volume integrals are especially important in physics for many
application, for exa, ple,to calculate flux densites.

4. Career Point Cares
Linear system
• A set of equations is called a system of equations. The
solutions of a system of equations must satisfy every
equation in the system. If all the equations in a system are
linear, the system is a system of linear equations, or a
linear system.

5. Career Point Cares
• A set of equations is called a
system of equations. The solutions
of a system of equations must
satisfy every equation in the
system. If all the equations in a
system are linear, the system is a
system of linear equations, or a
linear system.

6. Career Point Cares
2. The graphs are parallel
lines, so there is no
solution and the
solution set is ø. The
system is inconsistent
and the equations are
independent.

7. Career Point Cares
Substitution Method
In a system of two equations with two
variables, the substitution method involves
using one equation to find an expression for
one variable in terms of the other, and then
substituting into the other equation of the
system.

8. Career Point Cares
5.1 - 8
Example 1
SOLVING A SYSTEM
BY SUBSTITUTION
Solve the system.
3 2 11x y 
3x y  
(1)
(2)
Solution
Begin by solving one of the equations for one of the
variables. We solve equation (2) for y.
3x y   (2)
3y x  Add x.

9. Career Point Cares
5.1 - 9
Example 1
SOLVING A SYSTEM
BY SUBSTITUTION
Now replace y with x + 3 in equation (1), and solve
for x.
3 2 11x y  (1)
3 2( 113)xx   Let y = x + 3 in (1).
Note the careful
use of
parentheses.
3 2 6 11x x   Distributive property
5 6 11x   Combine terms.
5 5x  Subtract.
1x 

10. Career Point Cares
5.1 - 10
Replace x with 1 in equation (3) to obtain
y = 1 + 3 = 4. The solution of the system is the ordered
pair (1, 4). Check this solution in both equations (1) and
(2).
Example 1
SOLVING A SYSTEM
BY SUBSTITUTION
3 2 11x y  (1)
Check:
3yx   (2)
3( ) 2( 11 4) 1  ?
11 11 True
41 3   ?
3 3 True
Both check; the solution set is {(1, 4)}.

11. Career Point Cares
Applications
An integral is from where the flux of one quantity is equal to another quantity.
• Physical laws-
Examples: Gauss’s law in electrostatics, magnetism & gravity.
• Continuity equations-
In fluid dynamics, electromagnetism, quantum mechanics, relativity theory & a
number of other fields , there are continuity equations that describe the
conservation of mass, momentum, energy, probability, or other quantities.
• Inverse square laws-
Two examples are Gauss' law, which follows from the inverse-square Coulomb's
law, and Gauss' law for gravity, which follows from the inverse-square Newton's
law of universal gravitation. The derivation of the Gauss' law-type equation from
the inverse-square formulation (or vice versa) is exactly the same in both cases.

12. Career Point Cares
IN ELECTROSTATICS
Gauss's law may be expressed as:
where ΦE is the electric flux through a closed surface S enclosing any
volume V,
Q is the total charge enclosed within S,
and ε0 is the electric constant.
The electric flux ΦE is defined as a surface integral of the electric field:
where E is the electric field, dA is a vector representing
an infinitesimal element of area of the surface, and · represents the dot
product of two vectors.

13. Career Point Cares
IN MAGNETISM
The integral form of Gauss's law for magnetism states:
where S is any closed surface, and dA is a vector,
whose magnitude is the area of an infinitesimal piece of the surface S, and whose
direction is the outward-pointing surface normal.
The left-hand side of this equation is called the net flux of the magnetic field out
of the surface, and Gauss's law for magnetism states that it is always zero.
The integral and differential forms of Gauss's law for magnetism are
mathematically equivalent, due to the divergence theorem. That said, one or the
other might be more convenient to use in a particular computation.
The law in this form states that for each volume element in space, there are
exactly the same number of "magnetic field lines" entering and exiting the
volume.

14. Career Point Cares
IN GRAVITY
The integral form of Gauss's law for gravity states:
where (also written ) denotes a surface integral over a closed surface, ∂V is any
closed surface (the boundary of a closed volume V),dA is a vector, whose magnitude
is the area of an infinitesimal piece of the surface ∂V, and whose direction is the
outward-pointing surface normal, g is the gravitational field, G is the
universal gravitational constant, and M is the total mass enclosed within the surface
∂V. The left-hand side of this equation is called the flux of the gravitational field.
Note that according to the law it is always negative (or zero), and never positive.
This can be contrasted with Gauss's law for electricity, where the flux can be either
positive or negative. The difference is because charge can be either positive or
negative, while mass can only be positive.

15. Career Point Cares
MASS COUNTINUITY
Mass continuity (conservation of mass): The rate of change of fluid mass inside a
control volume must be equal to the net rate of fluid flow into the volume.
Physically, this statement requires that mass is neither created nor destroyed in the
control volume, and can be translated into the integral form of the continuity
equation:
Above, is the fluid density, u is the flow velocity vector, and t is time. The left-hand side
of the above expression contains a triple integral over the control volume, whereas
the right-hand side contains a surface integral over the surface of the control
volume.

16. Career Point Cares
CONSERVATION OF MOMENTUM
This equation applies Newton's second law of motion to the control volume, requiring
that any change in momentum of the air within a control volume be due to the net
flow of air into the volume and the action of external forces on the air within the
volume. In the integral formulation of this equation, body forces here are
represented by fbody, the body force per unit mass. Surface forces, such as viscous
forces, are represented by , the net force due to stresses on the control volume
surface.

17. Career Point Cares
Application of branch
• In this example, we are trying to solve for the forces located in the
beams. Since we do not have any initial conditions, we must solve for
variables, which is good. With variables we can change them at will
with very minimal hassal to observe the effects. We will apply the
loads only on the joints. You can see that we have applied
compression forces at all members, and labeled them for ease.
Applying the method of joints, we get these equations, labeled 1−6 for
the joints they are taken from:

18. Career Point Cares
conclusion
• systems of linear equations can have zero, one, or an
infinite number of solutions, depending on whether they
are consistent or inconsistent, and whether they are
dependent or independent. The first section will explain
these classifications and show how to solve systems of
linear equations by graphing.
• The second section will introduce a second method for
solving systems of linear equations--substitution.
Substitution is useful when one variable in an equation of
the system has a coefficient of 1or a coefficient that easily
divides the equation.

19. Career Point Cares
• If one of the variables has a coefficient of 1 , substitution is very useful
and easy to do. However, many systems of linear equations are not
quite so neat, and substitution can be difficult. The third section
introduces another method for solving systems of linear equations--the
Addition/Subtraction method.
• Systems of equations will reappear frequently in Algebra II. They will
be used in maximization and minimization problems, where solving by
graphing will become especially useful. Systems of equations also
appear in chemistry and physics; in fact, they are found in any situation
dealing with multiple variables and multiple constraints on them.

20. Career Point Cares
REFERENCES
• DAS H.K(2015)”Advance engineering mathematics”
s.chand publication,1st edition Pp.6,46,171
• N.P. Bali (2016)” engineering mathematics” Lakshmi
publication, 9th edition, PP 423,558
• B.V RAMANNA (2007)” engineering mathematics” 2nd
edition Pp 8.22
• D.K.K.Rewar (2004)” engineering mathematics” college
book house, New edition Pp;84
• Classroom maths notebook

21. Career Point Cares
Visited websites
• https://www.google.co.in/?gfe_rd=cr&ei=uXo1V5GTGabv8wfgi6zgB
A&gws_rd=ssl#safe=active&q=linear+system+
• https://www.google.co.in/?gfe_rd=cr&ei=uXo1V5GTGabv8wfgi6zgB
A&gws_rd=ssl#safe=active&q=vector+inregral
• https://www.google.co.in/?gfe_rd=cr&ei=uXo1V5GTGabv8wfgi6zgB
A&gws_rd=ssl#safe=active&q=matrix+system+application+in+civil
• http://www.ce.utexas.edu/prof/mckinney/ce311k/handouts/linear_equa
tions.pdf
• http://www.wil.pk.edu.pl/docs/kartyprogramowe/S1A/syllabus_B1.pd
f

22. Career Point Cares