lec20.ppt
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This document contains a series of tutorial problems related to matrices and linear algebra. Problem 1 asks to invert a 3x3 matrix. Problem 2 asks to write a vector as a linear combination of two other vectors. Problem 3 involves finding the inverse, trace, and determinant of related matrices. Problem 4 proves a property about powers of similar matrices. Problem 5 diagonalizes a 2x2 matrix and finds its eigenvalues and eigenvectors.
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![NPTEL – Physics – Mathematical Physics - 1
Lecture 20
Tutorials
1. Invert the following matrix,
1 1 1
𝐴 = (1 2 1)
1 1 3
2. Write 𝑏⃑⃗ as a linear combination of 𝑎⃑⃑⃑⃑1⃗ and 𝑎⃑⃑⃑⃑2⃗. 𝑏⃑⃗ = [3, 2], ⃑𝑎⃑⃑⃑1⃗ =
[4, 1], ⃑𝑎⃑⃑⃑2⃗ = [2, 5]
1 5
Ans: 𝐴⃗ = (𝑎⃑⃑⃑⃑1⃗, ⃑𝑎⃑⃑⃑2⃗) = [4
2] ; 𝐴−1 = [
1 5 − 2
18 −1 4
]
𝑦 = 𝐴−1𝑏⃑⃗ =
1
[ 18 −1 4 2
5 − 2 3
] [ ]
= 1
[11
18 5
]
𝑦1 =
18
, 𝑦2 =
18
11 5
[3
Page 16 of 17
Joint initiative of IITs and IISc – Funded by MHRD
2
] = ( ) [ ] + ( ) [ ]
3. 𝐴 = [
3 6
] and P = [ ]
4 − 2 1 2
3 4
a. Find 𝐵 = 𝑃−1𝐴𝑃
b. Verify 𝑇𝑟(𝐴) = 𝑇𝑟(𝐵), where Tr defines Trace of a matrix which means sum
of all the diagonal elements.
c. Verify 𝑑𝑒𝑡 (𝐵) = 𝑑𝑒𝑡 (𝐴).
−2 1
Ans: 1. 𝑃−1 = [3
2 2
− 1]
𝐵 = [3
−
1] [
3 6
] [ ]
−2 1
2 2
4 − 2 1 2
3 4
= [ 25 30
−27 − 15
]
2. 𝑇𝑟 (𝐴) = 𝑇𝑟 (𝐵) = 10.
3. 𝐷𝑒𝑡(𝐴) = 𝑑𝑒𝑡 (𝐵) = 30
11 4 5 2
18 1 18 5](https://image.slidesharecdn.com/lec20-231023100919-ae308e7b/85/lec20-ppt-1-320.jpg)
![NPTEL – Physics – Mathematical Physics - 1
4. Suppose B is similar to A, say 𝐵 = 𝑃−1𝐴𝑃. Prove that 𝐵𝑛 = 𝑃−1𝐴𝑛𝑃
and so 𝐵𝑛 is similar to 𝐴𝑛 for some interger n.
Ans: The proof follows by induction. The result holds for 𝑛 = 1 by hypothesis. For 𝑛 ≥
1, suppose the result holds for n-1 then
𝐵𝑛 = 𝐵𝐵𝑛−1 = (𝑃−1𝐴𝑃)(𝑃−1𝐴𝑛−1𝑃)
= 𝑃−1𝐴𝑛𝑃
5. Find the matrix 𝑋, that makes 𝐴 = [ ] diagonal.
3 1
2 2
Alsofind the eigenvalues and eigenvectors of A.
The eigenvectors, 𝑣 = [ ] and 𝑣 = [ ] and 𝜆 = 1 and 𝜆 = 4
1 1
1 1
−2 1 2
𝑋 = [ 1 1 ] ; 𝑋−1 = 1
[1
Page 17 of 17
Joint initiative of IITs and IISc – Funded by MHRD
−2 1
𝐷 = 𝑋−1𝐴𝑋 = [1
3 2
− 1]
1
0]
0 4](https://image.slidesharecdn.com/lec20-231023100919-ae308e7b/85/lec20-ppt-2-320.jpg)
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lec9.ppt
lec9.pptRai Saheb Bhanwar Singh College Nasrullaganj The document discusses linear independence and change of basis in vector spaces. It provides the following key points:
1) Two vectors u and v are linearly dependent if one is a multiple of the other, and independent otherwise.
2) Three vectors u, v, and w are linearly dependent if their coefficients in a linear combination equal 0, and independent otherwise.
3) A change of basis matrix P describes the transformation between two bases {e} and {f} of a vector space, where each vector in {f} is written as a linear combination of the vectors in {e}. The inverse of P transforms vectors back from {f} to {e}.
Maths-MS_Term2 (1).pdf
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lec15.ppt
lec15.pptRai Saheb Bhanwar Singh College Nasrullaganj The rank of a matrix is the maximum number of linearly independent rows. A matrix has rank 0 only if it is the zero matrix. The inverse of a matrix A, denoted A^-1, is defined such that A*A^-1 = I, the identity matrix. To calculate the inverse, one takes the transpose of the cofactor matrix and divides each element by the determinant of the original matrix.
BSC_COMPUTER _SCIENCE_UNIT-1_DISCRETE MATHEMATICS
BSC_COMPUTER _SCIENCE_UNIT-1_DISCRETE MATHEMATICSRai University This document discusses successive differentiation and provides examples of finding the nth derivative of common functions such as polynomials, exponentials, logarithms, trigonometric functions, and rational functions. Some key points covered include:
- The nth derivative of a function y with respect to x is denoted as d^n y/dx^n.
- Formulas are given for finding the nth derivative of functions such as x^m, e^ax, a^x, 1/(ax+b), (ax+b)^m, log(ax+b), sin(ax+b), and cos(ax+b).
- Examples demonstrate calculating specific high-order derivatives, such as the 10th derivative of
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BSC_Computer Science_Discrete Mathematics_Unit-IRai University This document discusses successive differentiation and provides examples of finding the nth derivative of common functions such as polynomials, exponentials, logarithms, trigonometric functions, and rational functions. Some key points covered include:
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CBSE Mathematics sample question paper with marking scheme
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More from Rai Saheb Bhanwar Singh College Nasrullaganj (20)
lec34.ppt
lec34.pptRai Saheb Bhanwar Singh College Nasrullaganj 1. The document discusses analytic functions of complex variables through examples. It defines analytic functions as those whose derivatives of all orders exist in the region of analyticity.
2. The Cauchy-Riemann equations are derived and their implications are explored, including that they imply the Laplace equation and orthogonality of level curves.
3. Several examples are worked through to determine if functions are analytic by checking if they satisfy the Cauchy-Riemann equations. The Cauchy-Riemann equations are also derived in polar coordinates.
lec33.ppt
lec33.pptRai Saheb Bhanwar Singh College Nasrullaganj The Cauchy Riemann (CR) conditions provide a necessary and sufficient condition for a function f(z) = u(x, y) + iv(x, y) to be analytic in a region. The CR conditions require that the partial derivatives of u and v satisfy ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x. If a function satisfies these conditions at all points in a region, then it is analytic in that region. The document proves this using cases where ∆y = 0 and ∆x = 0, showing the derivatives must be equal. Examples are provided to demonstrate checking functions for analytic
lec31.ppt
lec31.pptRai Saheb Bhanwar Singh College Nasrullaganj Complex numbers allow solutions to equations like x2 + 1 = 0 by extending real numbers to include imaginary numbers. A complex number z is defined as z = x + iy, where x and y are real numbers and i is the imaginary unit equal to √-1. Complex numbers can be added and multiplied following specific rules, such as z1 + z2 = (x1 + x2) + i(y1 + y2) for addition and z1z2 = (x1x2 - y1y2) + i(y1x2 + x1y2) for multiplication. The inverse of a complex number z is calculated as z-1 = (x/(x2+y
lec42.ppt
lec42.pptRai Saheb Bhanwar Singh College Nasrullaganj 1) The document discusses examples of calculating the Jacobian of transformations. It defines the Jacobian as the determinant of the partial derivatives of the transformed coordinates.
2) It then discusses Möbius transformations, which are fractional linear transformations of the form (az+b)/(cz+d). The Jacobian of a Möbius transformation depends only on z.
3) Several examples are given of using Möbius transformations to map one geometric region to another, such as mapping a circle to a line.
lec41.ppt
lec41.pptRai Saheb Bhanwar Singh College Nasrullaganj This document discusses conformal mapping, which maps curves and regions in such a way that preserves angles and their directions. It provides examples of conformal mappings:
1) The mapping w = ez maps a vertical line in the z-plane to a circle in the w-plane, with the phase angle increasing along the circle.
2) The mapping ω = eiθ0(z-z0)/(z-z0) maps an area in the upper half z-plane to the interior of a unit circle in the ω-plane. Points on the x-axis in z are mapped to the boundary of the circle.
lec39.ppt
lec39.pptRai Saheb Bhanwar Singh College Nasrullaganj The document discusses Taylor and Laurent series expansions. It provides examples of using these expansions to represent functions around points.
Taylor series provides a power series representation of an analytic function around a point. Laurent series allows representing functions in annular regions, including points where the function is not analytic, using both positive and negative powers of (z - z0). Examples show deducing Laurent series expansions for simple functions like z4 and 1/z4 around various points, and evaluating coefficients via contour integrals and the residue theorem. The document also gives an example of using a contour integral to compute a Greens function in many-particle physics.
lec38.ppt
lec38.pptRai Saheb Bhanwar Singh College Nasrullaganj 1) Jordan's lemma is used to convert real integrals over the infinite real axis into complex integrals over a contour enclosing the real axis in the complex plane.
2) Several examples are provided of using residues and Jordan's lemma to evaluate definite integrals over the real line or infinite intervals that involve functions with poles, including integrals of x^2, sin(x)/x, 1/(x^2+a^2)^2, and sin(x)/(x(x^2+a^2)).
3) The technique involves closing the contour with a semicircle at infinity where the integral over the semicircle goes to zero by Jordan's lemma, leaving the original integral equal to the residue theorem applied to the
lec37.ppt
lec37.pptRai Saheb Bhanwar Singh College Nasrullaganj 1) The document discusses evaluating contour integrals using the residue theorem. It provides examples of calculating residues and evaluating integrals where the contour encloses poles.
2) The residue of a function f(z) at a pole z=a is the coefficient of the (z-a)^-1 term in the Laurent series expansion of f(z) about z=a.
3) According to the residue theorem, the value of a contour integral of a function along a closed loop is equal to 2πi times the sum of the residues of the function enclosed by the contour.
lec23.ppt
lec23.pptRai Saheb Bhanwar Singh College Nasrullaganj 1) The document discusses representation of the Dirac delta function in cylindrical and spherical coordinate systems. It shows that δ(r - r') = δ(ρ - ρ')δ(φ - φ')δ(z - z')/ρ in cylindrical coordinates and δ(r - r') = δ(r - r')δ(θ - θ')δ(φ - φ')/r^2 in spherical coordinates.
2) It also derives the important relation ∇^2(1/r) = -4πδ(r) and shows its application to the Laplace equation for electrostatic potential.
3) The completeness of eigenfunctions of harmonic oscillators and Legend
lec21.ppt
lec21.pptRai Saheb Bhanwar Singh College Nasrullaganj 1. The Dirac delta function is an important concept in quantum mechanics and electrodynamics that describes an impulse or large force acting over a very short time interval.
2. The key properties of the Dirac delta function are that it is equal to infinity at a single point and zero everywhere else, and that the integral of the function over its entire range is equal to one.
3. The Dirac delta function can be used to find the value of an arbitrary function f(x) at a specific point a, as the integral of f(x) multiplied by the Dirac delta function over all x is equal to f(a).
lec19.ppt
lec19.pptRai Saheb Bhanwar Singh College Nasrullaganj This document discusses finding the eigenvalues and eigenfunctions of a spin-1/2 particle pointing along an arbitrary direction. It shows that the eigenvalue equation reduces to a set of two linear, homogeneous equations. The eigenvalues are found to be ±1/2, and the corresponding eigenvectors are written in terms of the direction angles θ and Φ. As an example, it shows that for a spin oriented along the z-axis, the eigenvectors reduce to simple forms as expected for a spin-1/2 particle. It also introduces the Gauss elimination method for numerically solving systems of linear equations that arise in eigenvalue problems.
lec18.ppt
lec18.pptRai Saheb Bhanwar Singh College Nasrullaganj This document discusses solving a mass-spring system as an eigenvalue problem. It:
1) Sets up differential equations to model the displacements of two masses connected by springs.
2) Transforms the coupled differential equations into a matrix eigenvalue equation.
3) Solves the eigenvalue equation to obtain the frequencies of oscillation for the two masses.
4) Combines the eigenvectors with complex exponential functions to obtain general solutions for the displacements of each mass over time.
lec17.ppt
lec17.pptRai Saheb Bhanwar Singh College Nasrullaganj This document discusses linear transformations and matrices. It introduces how linear transformations on physical quantities are usually described by matrices, where a column vector u representing a physical quantity is transformed into another column vector Au by a transformation matrix A. As an example, it discusses orthogonal transformations, where the transformation matrix A is orthogonal. It proves that for an orthogonal transformation, the inner product of two vectors remains invariant. It also discusses properties of other types of matrices like Hermitian, skew-Hermitian and unitary matrices.
lec16.ppt
lec16.pptRai Saheb Bhanwar Singh College Nasrullaganj This document discusses properties of symmetric, skew-symmetric, and orthogonal matrices. It defines each type of matrix and provides examples. Key points include:
- Symmetric matrices have Aij = Aji for all i and j. Skew-symmetric matrices have Aij = -Aji. Orthogonal matrices satisfy AT = A-1.
- The eigenvalues of symmetric matrices are always real. The eigenvalues of skew-symmetric matrices are either zero or purely imaginary.
- Any real square matrix can be written as the sum of a symmetric matrix and skew-symmetric matrix.
lec30.ppt
lec30.pptRai Saheb Bhanwar Singh College Nasrullaganj 1) The document discusses calculating the moment of inertia tensor for a cylinder with radius R and height H. It is shown that the only non-zero components of the inertia tensor are Ixx = (3MH + 4MR2)/12, Iyy = Ixx, and Izz = MR2/2.
2) Equations for velocity, acceleration, and the Christoffel symbols in an arbitrary coordinate system are presented. Expressions for calculating acceleration in cylindrical coordinates using the metric tensor and Christoffel symbols are given.
lec28.ppt
lec28.pptRai Saheb Bhanwar Singh College Nasrullaganj Tensors obey algebraic properties including addition, multiplication, contraction, and symmetrization. Addition of tensors combines their components. Multiplication of tensors combines their indices and ranks to form a new tensor. Contraction sets equal a covariant and contravariant index, reducing the tensor's rank. Symmetric tensors do not change sign under index interchange, while antisymmetric tensors change sign.
lec27.ppt
lec27.pptRai Saheb Bhanwar Singh College Nasrullaganj 1) The document discusses tensors with multiple indices and the cross product of two vectors A and B. The components of the cross product vector C are given by Ai Bj - Aj Bi.
2) It describes how tensor components transform between coordinate systems using transformations of partial derivatives. The transformation property for cross products is derived.
3) Tensors are defined by their rank, with the number of covariant and contravariant indices specifying a tensor's rank. Vectors have a rank of 1. Examples calculate tensor components in different coordinate systems.
lec26.ppt
lec26.pptRai Saheb Bhanwar Singh College Nasrullaganj 1) There are two types of vectors - contravariant vectors whose components transform according to Equation 1, and covariant vectors whose components transform according to Equation 2.
2) The dot product of two contravariant or two covariant vectors is not independent of the coordinate system.
3) The dot product of a contravariant and a covariant vector is independent of the coordinate system.
lec25.ppt
lec25.pptRai Saheb Bhanwar Singh College Nasrullaganj The document discusses the transformation properties of vectors between two coordinate systems. It states that if a vector has components (x1, x2, x3) in one coordinate system and (x̅1, x̅2, x̅3) in another, there is a relation such that x̅i = ai1x1 + ai2x2 + ai3x3, where the aij are the components of the transformation matrix. This relation can be written compactly as x̅i = aijxj, using Einstein summation convention. The document then presents two theorems establishing that a set of functions can represent a tensor if their inner product transforms as a tensor under coordinate transformations.
lec2.ppt
lec2.pptRai Saheb Bhanwar Singh College Nasrullaganj 1) The document discusses the gradient operator and provides examples of calculating the electric field and gradient of functions.
2) It defines the gradient operator mathematically and provides an example of calculating the gradient of a scalar function.
3) Examples are given of using the gradient operator to calculate the electric field from a potential and the gradient of 1/r.
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• Kansas: 1 - This is 2.7% of all KS cases.
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INTERNATIONAL SPREAD (As of 4/2/2025)
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‒Chihuahua, Mexico: 844 (+58) cases, 3 hospitalizations, 1 fatality
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Physical Properties
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Non-metals: Dull appearance (except iodine, which is shiny).
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Non-metals: Generally do not react with water.
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Displacement Reaction
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Aluminum: Used in aircraft, utensils.
Copper: Electrical wires.
Gold and Silver: Jewelry.
Zinc: Coating iron to prevent rusting (galvanization).
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Nitrogen: Fertilizers.
Chlorine: Water purification.
Carbon: Fuel (coal), steel-making (coke).
Iodine: Medicines.
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lec20.ppt
- 1. NPTEL – Physics – Mathematical Physics - 1 Lecture 20 Tutorials 1. Invert the following matrix, 1 1 1 𝐴 = (1 2 1) 1 1 3 2. Write 𝑏⃑⃗ as a linear combination of 𝑎⃑⃑⃑⃑1⃗ and 𝑎⃑⃑⃑⃑2⃗. 𝑏⃑⃗ = [3, 2], ⃑𝑎⃑⃑⃑1⃗ = [4, 1], ⃑𝑎⃑⃑⃑2⃗ = [2, 5] 1 5 Ans: 𝐴⃗ = (𝑎⃑⃑⃑⃑1⃗, ⃑𝑎⃑⃑⃑2⃗) = [4 2] ; 𝐴−1 = [ 1 5 − 2 18 −1 4 ] 𝑦 = 𝐴−1𝑏⃑⃗ = 1 [ 18 −1 4 2 5 − 2 3 ] [ ] = 1 [11 18 5 ] 𝑦1 = 18 , 𝑦2 = 18 11 5 [3 Page 16 of 17 Joint initiative of IITs and IISc – Funded by MHRD 2 ] = ( ) [ ] + ( ) [ ] 3. 𝐴 = [ 3 6 ] and P = [ ] 4 − 2 1 2 3 4 a. Find 𝐵 = 𝑃−1𝐴𝑃 b. Verify 𝑇𝑟(𝐴) = 𝑇𝑟(𝐵), where Tr defines Trace of a matrix which means sum of all the diagonal elements. c. Verify 𝑑𝑒𝑡 (𝐵) = 𝑑𝑒𝑡 (𝐴). −2 1 Ans: 1. 𝑃−1 = [3 2 2 − 1] 𝐵 = [3 − 1] [ 3 6 ] [ ] −2 1 2 2 4 − 2 1 2 3 4 = [ 25 30 −27 − 15 ] 2. 𝑇𝑟 (𝐴) = 𝑇𝑟 (𝐵) = 10. 3. 𝐷𝑒𝑡(𝐴) = 𝑑𝑒𝑡 (𝐵) = 30 11 4 5 2 18 1 18 5
- 2. NPTEL – Physics – Mathematical Physics - 1 4. Suppose B is similar to A, say 𝐵 = 𝑃−1𝐴𝑃. Prove that 𝐵𝑛 = 𝑃−1𝐴𝑛𝑃 and so 𝐵𝑛 is similar to 𝐴𝑛 for some interger n. Ans: The proof follows by induction. The result holds for 𝑛 = 1 by hypothesis. For 𝑛 ≥ 1, suppose the result holds for n-1 then 𝐵𝑛 = 𝐵𝐵𝑛−1 = (𝑃−1𝐴𝑃)(𝑃−1𝐴𝑛−1𝑃) = 𝑃−1𝐴𝑛𝑃 5. Find the matrix 𝑋, that makes 𝐴 = [ ] diagonal. 3 1 2 2 Alsofind the eigenvalues and eigenvectors of A. The eigenvectors, 𝑣 = [ ] and 𝑣 = [ ] and 𝜆 = 1 and 𝜆 = 4 1 1 1 1 −2 1 2 𝑋 = [ 1 1 ] ; 𝑋−1 = 1 [1 Page 17 of 17 Joint initiative of IITs and IISc – Funded by MHRD −2 1 𝐷 = 𝑋−1𝐴𝑋 = [1 3 2 − 1] 1 0] 0 4