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**

**Course Outline**

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.

MTH622 Handouts pdf

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### MTH622 HANDOUTS

**MTH622: Vectors and Classical Mechanics**

**Scalar Point Function**

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.

#### Chasle’s Theorem

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.