PHY 103 N Week 1.

Lecture 1. ( PLEASE ALSO READ COMMENTS ON THE FIRST WELCOME POST).
In this week we first learnt about scalars, vectors and their transformation properties. Typically a quantity is a vector under a set of coordinate transformations provided its components transforms in a specific way under that set of transformations.The components in the transformed coordinate system are functions of the components in the old coordinate system. Like in rotations and inversions. We will be mostly concerned with vectors under rotations.
*Actually the word "direction" is dependent on which coordinate system one uses. The transformation law is a more accurate specification of the 'direction" of a vector which is independent of coordinate system in use.
* Tensors are objects with more components and may be described as products of vectors. Like rank 2 tensor is like a product of two vectors having 9 components in three dimensions all of which may not be independent. Scalars are rank 0 and vectors are rank 1 tensors. Tensors transform like products of vectors. So a rank 2 tensor under rotations will transform as the product of two rotation matrices under rotations.
* A pesudovector transforms just like a vector under rotations but with a relative minus sign to vectors under inversions. Simmilarly for pesduscalars and pseudo tensors.
Lecture 2
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Introduced scalar fields and vector fields as scalar and vector functions of the co-ordinates. A scalar transforms as φ'(x', y', z')=φ ( x, y, z). Note that the function on RHS is different from the function on the LHS but the old function evaluated in the old coordinates is equal to the numerical value of the new function in the new co-ordnate system.A vector field has three components ( in 3 dimensions) each of which transforms like a scalar field.
The ∇ or the gradient operator has been introduced. This is a vector operator
such that under rotations its components which are ∇ = i ∂ / ∂ x + j ∂ / ∂ y + k ∂ / ∂ z which transform like a vector . Note that although it has no magnitude or direction its components transform like a vector under rotations.
The gradient of a scalar field ∇ φ ( x,y,z) is a vector field. It provides the direction of maximal change in φ. Constant φ surfaces are called level surfaces and the direction of grad φ is always normal to the level surface.
Lecture 3.
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Introduced the divergence of a vector field ∇ . v ( x, y, z) given by the dot product of the grad operator with a vector field. The divergence of a vector field provides a measure of the "spread" or relative increase or decrease in "magnitude" around that point. If v is a fluid flow velocity vector then div v positive signifies a "tap" and negative signifies a "sink" and zero signifies a "laminar flow".
Introduced the Curl of a vector field ∇ x v( x, y, z) given by the cross product of the grad operator with a vector field v. The result is another vector field. The curl of a vector field at a point provides a measure of how much the direction of the vector field curls or wraps around the point in question. Curl can be non zero also due to the relative magnitudes being different in different
points around the point in question. For a fluid flow the non zero curl is the regions where a "whirlpool" or a "vortex" appears.
We also learnt about other products and vector identities with the grad operator. Please NOTE that some of the vector identities are very different for the grad operator. This is because the grad apart from being a vector is also a differential operator.
We also introduced the scalar operator ∇ ² which is the sum of the second partial derivaties with respect to x, y, z. The Laplacian can act both on scalar and vector fields. On a vector field v the Laplacian acts on each component of the vector field. ( remember each component of a vector field is a scalar field itself. so it is the sum of the Laplacian acting on the 3 scalar fields which are the components of the vector field v.