Engineering Mathematics III MCQs and notes | diplomachakhazana

 

UNIT I: Linear systems of equations:

Rank-Echelon form-Normal form – Solution of linear systems – Gauss elimination – Gauss Jordon- Gauss Jacobi and Gauss-Seidel methods. Applications: Finding the current in electrical circuits.

UNIT II: Eigenvalues – Eigenvectors and Quadratic forms:

Eigenvalues – Eigenvectors– Properties – Cayley-Hamilton theorem Inverse and powers of a matrix by using Cayley-Hamilton theorem- Diagonalization- Quadratic forms- Reduction of quadratic form to canonical form – Rank – Positive, negative and semidefinite – Index – Signature. Applications: Free vibration of a two-mass system.

UNIT III: Multiple integrals:

Curve tracing: Cartesian, Polar, and Parametric forms. Multiple integrals: Double and triple integrals – Change of variables –Change of order of integration. Applications: Finding Areas and Volumes.

UNIT IV: Special functions:

Beta and Gamma functions- Properties – Relation between Beta and Gamma functions- Evaluation of improper integrals.
Applications: Evaluation of integrals.

UNIT V: Vector Differentiation:

Gradient- Divergence- Curl – Laplacian and second-order operators -Vector identities. Applications: Equation of continuity, potential surfaces

UNIT VI: Vector Integration:

Line integral – Work is done – Potential function – Area- Surface and volume integrals Vector integral theorems: Greens, Stokes, and Gauss Divergence theorems (without proof) and related problems.
Applications: Work is done, Force.

• Formation of differential equation

  Order = no. of essential arbitrary constants

  (i) Identify essential arbitrary constants

  (ii) Differentiate till required order

     (iii) Now eliminate arbitrary constants from the equation of curve or any other equation obtained from it.

• A solution of a differential equation is an equation which contains as many arbitrary constants as the order of the differential equation and is termed as the general solution of the differential equation.

• The solutions obtained by giving particular values to the arbitrary constants in the general solution are termed as the particular solutions.

• The general solution of a differential equation of the nth order must contain n and only n independent arbitrary constants. 

• A separable differential equation can be expressed as the product of a function of x and a function of y i.e.it can be expressed in the form

    f(x)dx + g(y)dy = 0.

• The solution of such an equation is given by

    ∫f(x)dx + ∫g(y)dy = c, where ‘c’ is the arbitrary constant.

• If the equation is of the form:
  
  
  
  To solve this, just substitute t = ax + by + c. Then the equation reduces to separable type in the variable t and x which can be easily solved.

• At times, transformation to the polar coordinates facilitates the separation of variables. In that case, it is advisable to remember the following differentials:

• If x = r cos θ and y = r sin θ then

                    1. x dx + y dy = r dr

                    2. x dy – y dx = r² dθ

• If x = r sec θ and y = r tan θ then

                    1. x dx – y dy = r dr

                    2. x dy – y dx = r² sec θ dθ

• A differential equation of the form
  
  
  
  where both f(x,y) and φ(x,y) are homogeneous functions of x and y and of the same degree is called homogeneous. This equation can also be reduced to the form
  
  
  
  and is solved by putting y = vx so that the differential equation is transformed into an equation with variable separable.

• Equation of the form
  
  
  
  can be reduced to a homogeneous form by changing the variable x, y to X, Y by writing x = X + h and y = Y + k, where h and k are constants to be chosen in such a way that

          ah + bk + c = 0

          and Ah + Bk + C = 0.    

• A differential equation is said to be linear if the dependent variable and all its differential coefficients occur in degree one only and are never multiplied together.

• Linear differential equations of first order first degree

          1.  

       where P & Q are functions of x

       I.F. = e ^∫Pdx

      Solution is e ^∫Pdx = ∫ Qe ^∫Pdx dx + c

         2.  

       where P1, Q1 are functions of y alone or constants

        I.F. = e ∫P1dy

       Solution is e ∫P1dy = ∫ Q1e ∫P1dy dy + c

         3. Reducible to linear (Bernoulli’s equation)

             

         P and Q are functions of x are functions of x.

         The equation can be reduced to linear equation by dividing by yⁿ & then substituting y ^-n+1 = Z.

• Exact differentials to be remembered:      

         

• Exact differential equation: A differential equation M(x, y)dx + N(x, y) dy = 0 is called an exact differential equation if there exists a function u such that du = Mdx + Ndy.

• The above differential equation Mdx + Ndy = 0 is termed to be exact if ∂M/∂y = ∂N/∂x. Its solution hence is given by ∫M dx + ∫N dy = c, where in the first integral, i.e. in M, y is considered as a constant and in N, only those terms which are independent of x are considered.

• Clairaut form of differential equation: The differential equation of the form y = Px + f(p), where P = dy/dx. The solution of this equation is obtained by replacing P by C.

• Physical Applications of Differential Equations:

• Mixture problems

• Statistical Applications

• Geometrical Applications

• Trajectories 

• Some Results on Tangents and Normals:

 

As shown in the adjacent figure, PT is defined as the length of the tangent while PN is defined as the length of the normal. Moreover, TM is defined as the sub-tangent while MN is defined as the length of the sub-normal. 

     

 

1. The length of the tangent = PT = y √[1 + (dx/dy)²]

2. The length of the normal = PN = y √[1 + (dy/dx)²]

3. Length of sub-tangent = TM = |y dx/dy|

4. Length of sub-normal = TM = |y dy/dx|

• The equation of the tangent at P(x, y) to the curve y = f(x) is

           Y – y =  dy/dx .(X-x)

• The equation of the normal at point P(x, y) to the curve y = f(x) is Y – y =  [-1/ (dy/dx) ].(X – x )

• Suppose we have a family of plane curves φ(x, y, a) = 0, depending on a single parameter ‘a’. A curve making at each of its points a fixed angle α with the curve of the family passing through that point is called an isogonal trajectory of that family. In case, α = π/2, then it is termed to be an orthogonal trajectory.

• How to find the orthogonal trajectory:

    Suppose the differential equation of the given family of curves is of the form F(x, y , y’) = 0

    The differential equation of the orthogonal trajectories is of the form F(x, y, -1/y’) = 0

Click the BUTTON to download mcq

Check out our page for more updates