Basic Mathematical Concepts Physics Notes

Basic Mathematical Concepts Notes

Physics deals with the study of nature. It tells about the fundamental laws of nature that govern the various physical phenomena in the world. Measurement of such a physical quantity involves the comparison with a basic and internationally accepted reference standard called UNIT. A physical quantity is expressed by a number (or numerical measure) accompanied by a unit.

For example, if the mass of an object is 7 kg. Then, it means that the mass of the object is 7 times the unit (i.e., 7 times of 1 kg).

But, there are some physical quantities that can not be completely expressed by the number and the unit. For example, let us consider that a boy Abhimanyu moves from a reference point O to cover a distance of 100 m. The distance between the point O to 100 m can be in infinite directions. The figure shows some of the displacements by Abhimanyu as P₁, P₂, P₃,…, etc.
Basic Mathematical Concepts Physics Notes 1
It is apparent that without knowing about the direction of motion of Abhimanyu, we can not say anything about the final position of Abhimanyu.

All the measurable physical quantities can be divided into two classes, namely: i. Scalar quantities 2. Vector quantities

→ Scalar Quantities: Scalars are quantities that are fully described by a magnitude (or numerical value) only.
Example: Distance, mass, density, etc.

→ Vector quantities: The physical quantities which require both magnitude and direction for their description are called vector quantities.

→ Kinds of vectors:

Axial vectors: Axial vectors are used to describe rotational motion. These are those which represent rotational effect and act along the axis of rotation according to the right hand screw rule example: angular velocity (\(\overrightarrow{(\omega)}\)), angular acceleration (\(\overrightarrow{(α)}\)), etc.
Polar vectors: Polar vectors describe translation motion and have a starting point. The direction of the polar vector remains unchanged irrespective of the coordinate system chosen. Example: displacement, force, etc.

→ Some important definitions regarding vectors:

Equal vector: Two vectors are said to be equal if they have the same magnitude and direction.
Opposition negative vector: A negative or opposite vector is defined as a vector having the same length but drawn in opposite direction.
Unequal vector: Two vectors are said to be unequal vectors; if they have equal magnitude but opposite direction or they have equal direction but opposite magnitudes or they have unequal magnitudes and direction.
Zero vector: A vector whose magnitude is zero is called a zero vector.
Unit vector: A vector that has a magnitude of one.

→ Addition of vectors

Triangle law of vector addition: It states that when two vectors are represented by two sides of a triangle in magnitude and direction taken in the same order then the third side of that triangle represents in magnitude and direction the resultant of the vectors.
Parallelogram law of vector addition: It states that if two vectors are considered to be the adjacent sides of a parallelogram, then the resultant of two vectors is given by the vector which is a diagonal passing through the point of contact of two vectors.
Polygon law of vector addition: This law is used to add more than two vectors. It states that if two or more vectors are represented by the adjacent sides of a polygon, taken in the same order both in magnitude and direction, then the result is given by the closing side of the polygon taken in opposite order both in magnitude and direction.

→ Subtraction of vectors: The process of subtracting one vector from the other is equivalent to adding, vectorially, the negative of the vector to be subtracted.

→ Product of vectors:

Scalar product or dot product: The scalar product or the dot product is the product of two vectors which are given magnitude only. The dot product is written using a central dot (.).
\(\overrightarrow{\mathrm{A}}\). \(\overrightarrow{\mathrm{B}}\) =|\(\overrightarrow{\mathrm{A}}\)||\(\overrightarrow{\mathrm{B}}\)| cosθ = ABcosθ
Cross product or vector product: The vector product or cross product is the product of two vectors which give a physical quantity which has both magnitude and direction.
or \(\overrightarrow{\mathrm{A}}\) x \(\overrightarrow{\mathrm{B}}\) = AB sinθ.n̂

→ Differential calculus: It is a subfield of calculus concerned with the study of the rates at which quantities change.

→ Integral calculus: A branch of mathematics concerned with the theory and applications of integrals and integration.

→ Scalar Quantity: Those quantities which required only magnitude for their description are called scalar quantities.

→ Vector quantity: Those physical quantities which are described both by direction and magnitude are called vector quantities.

→ Tensor quantity: Those quantities which are note even clearly defined by magnitude and direction are called tensor quantities.

→ Axial vectors: A quantity that transforms like a vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection.

→ Polar vectors: Those vectors whose initial point is definite are “ailed polar vectors.

→ Zero vector: Those vectors whose magnitude is zero.

→ Unit vector: Those vectors whose magnitude is one.

→ Position vector: Vector representing the position or point or any object is called the position vector.