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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)

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CONTENTSIntroduction Vector integral• Line integral• Surface integral• Volume integral

Linear system• Realworld application• Applications of branch• Conclusion• Referance

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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.

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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.

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• 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.

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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.

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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.

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Example 1SOLVING 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.

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Example 1SOLVING 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

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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 1SOLVING 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)}.

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ApplicationsAn 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.

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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.

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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.

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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.

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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. 

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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.

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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 e ects. We will apply the ffloads 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:

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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.

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• 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.

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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

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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=uXo1V5GTGabv8wfgi6zgBA&gws_rd=ssl#safe=active&q=vector+inregral

• https://www.google.co.in/?gfe_rd=cr&ei=uXo1V5GTGabv8wfgi6zgBA&gws_rd=ssl#safe=active&q=matrix+system+application+in+civil

• http://www.ce.utexas.edu/prof/mckinney/ce311k/handouts/linear_equations.pdf

• http://www.wil.pk.edu.pl/docs/kartyprogramowe/S1A/syllabus_B1.pdf

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