MTH622 Handouts pdf | Vectors and Classical Mechanics Notes (pdf)
Table of Contents
MTH622: Vectors and Classical Mechanics
A vector is an object that has both magnitude and direction. Geometrically, we can imagine a vector as a directed line whose length is the magnitude of the vector and with an arrow indicating the direction. MTH622 Handouts pdf
MTH622 Handouts pdf
Course Category: Mathematics
Scalar and Vector Fields, Properties of the Gradient, Directional Derivative, Geometrical interpretation of Gradient, Divergence of a vector Point Function, Properties of the Divergence, Laplacian, Curl of a vector Point Function, Properties of the Curl, Vector Identities, the line integral, Line Integral dependent on Path, Line Integral Independence of Path, Surface integral, Volume integral, Divergence theorem, Stokes’ Theorem, Simply and Multiply Connected Regions, Green’s Theorem in the Plane, Green””s theorem in the plane in vector notation, Green’s first identity, Green’s second identity.
Introduction to Classical Mechanics, newton’s law, and Rectangular components of velocity & acceleration. Introduction to tangential and normal components of velocity & acceleration, Curvature and radius of curvature, Newton””s first and second laws, The third law and law of Conservation of momentum, Validity of Newton’s law, Kinetic energy, Introduction to work – Theorem, Conservative force field, Non-conservative Force field, Introduction to simple harmonic motion and oscillator, Introduction to the damped harmonic oscillator, Euler’s theorem – derivation, Chasle’s theorem, Kinematics of a system of particles(space, time & matter), The concept of Rectilinear motion of particles Uniform rectilinear motion, uniformly accelerated rectilinear motion, The concept of curvilinear motion of particles,
Introduction to Projectile, the motion of a projectile, Conservation of energy for a system of particles, Introduction to impulse – Derivation, Introduction to torque, Introduction to rigid bodies and elastic bodies, Definition of Moment of Inertia and product of inertia, Radius of Gyration, Principal axes for the inertia matrix, Introduction to the Dynamics of a system of particles, Introduction to the center of mass and Linear momentum, Law of conservation of momentum for multiple particles, Angular momentum – derivation, Angular momentum in case of a continuous distribution of mass, Law of conservation of angular momentum, Kinetic Energy of a system about principal axes – Derivation, Moment of Inertia of a rigid body about a given line, Ellipsoid of Inertia, Moment of Inertia in Tensor Notation, Existence of principle axes theorem and its results, Determination of principal axes of other two when one is known, Rotational Kinetic Energy, Introduction to Special Moments of Inertia,
Solid Circular disk or Cylinder – Derivation, Hoop or cylindrical shell – Derivation, Solid Sphere – Derivation, Hollow sphere – Derivation, Rectangular Plate – Derivation, Thin Rod – Derivation, Planer motion of a rigid body, The compound pendulum, Equation of motion for the compound pendulum, Principal of linear momentum, Principal of angular momentum, Conservation of energy theorem in plane rotational motion, Work energy relation in case of rotational motion, General Motion of a Rigid Body in Space, Euler’s dynamical equations, Invariable direction and invariable Plane, Euler’s dynamical equations of a rigid body and its applications, Euler’s equation of Constrained rotation of rigid body about a fixed axis, Euler’s equation in case of free rotation of rigid body, Free Rotation of a Rigid Body with an Axis of Symmetry, Introduction to Euler’s Angles, Euler’s angles and rigid body motion, Angular velocity and kinetic energy in terms of Euler’s angles, Equimomental systems – Theorem.
MTH622 Handouts pdf
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If to each point (𝑥, 𝑦, 𝑧) of a region R in space there corresponds a scalar 𝜑(𝑥, 𝑦, 𝑧), then 𝜑 is called a scalar point function in R. Scalar field is a function defined on space whose value at each point is a scalar quantity The set of all values of scalar point function 𝜑 in R together forms a Scalar field.
Vector Point Function
If to each point (𝑥, 𝑦, 𝑧) of a region R in space there exists a unique vector 𝐴⃗ (𝑥, 𝑦, 𝑧), then 𝐴⃗ is called a vector point function in R. A function of a space whose value at each point is a vector quantity is called a vector field. Mathematically, we can write it as 𝐴⃗=𝐴⃗ (𝑥, 𝑦, 𝑧) =𝐴1 (𝑥, 𝑦, 𝑧)+ 𝐴2 (𝑥, 𝑦, 𝑧) +𝐴3 (𝑥, 𝑦, 𝑧) The set of all values of 𝐴⃗ in R constitute a vector field.
In kinematics, the Chasles theorem or Mozzi-Chasles theorem states that the most general displacement of a rigid body can be produced by displacement along a straight line (called its helical axis or Mozzi axis) followed by (or preceded by) a rotation about an axis parallel to that straight line.