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Basic Mathematics(Pre-requisite)

Differential calculus

Do you know what a physical quantity is? A physical quantity is any physical property that can be expressed in numbers. For example, time is a physical quantity as it can be expressed in numbers, but beauty is not as it cannot be expressed in numbers.

Scalar Quantities

  • If a physical quantity can be completely described only by its magnitude, then it is a scalar quantity. To measure the mass of an object, we only have to know how much matter is present in the object. Therefore, mass of an object is a physical quantity that only requires magnitude to be expressed. Therefore, we say that mass is a scalar quantity.

  • Some more examples of scalar quantities are time, area, volume, and energy.

  • We can add scalar quantities by simple arithmetic means.

  • It is difficult to plot scalar quantities on a graph.

Vector Quantities

  • There are some physical quantities that cannot be completely described only by their magnitudes. These physical quantities require direction along with magnitude. For example, if we consider force, then along with the magnitude of the force, we also have to know the direction along which the force is applied. Therefore, to describe a force, we require both its magnitude and direction. This type of physical quantity is called a vector quantity.

Therefore, we can define vector quantity as the physical quantity that requires both magnitude and direction to be described.

  • Some examples of vector quantities are velocity, force, weight, and displacement.

  • Vector quantities cannot be added or subtracted by simple arithmetic means.

  • Vector quantities can easily be plotted on a graph.

Scalars v/s Vectors

Scalars

Vectors

A scalar quantity has only magnitude.

A vector quantity has both magnitude and direction.

Scalars can be added, subtracted, multiplied, and divided just as ordinary numbers i.e., scalars are subjected to simple arithmetic operations.

Vectors cannot be added, subtracted, and multiplied following simple arithmetic laws. Arithmetic division of vectors is not possible at all.

Example: mass, volume, time, distance, speed, work, temperature

Example: displacement, velocity, acceleration, force

Position Vector

Position vector of a point in a coordinate system is the straight line that joins the origin and the point.

Magnitude of the vector is the length of the straight line and its direction is along the angle θ from the positive x-axis.

Displacement Vector

Displacement vector is the straight line joining the initial and final positions.

Equality of Vectors

Two vectors  and are said to be equal, if and only if they have the same magnitude and the same direction.

Calculus

Calculus is basically a way of calculating rate of changes (similar to slopes, but called derivatives in calculus), areas, volumes, and surface areas (for starters).

It’s easy to calculate these kinds of things with algebra and geometry if the shapes you’re interested in are simple. For example, if you have a straight line you can calculate the slope easily. But if you want to know the slope at an arbitrary point (any random point) on the graph of some function like x-squared or some other polynomial, then you would need to use calculus. In this case, calculus gives you a way of “zooming in” on the point you’re interested in to find the slope exactly at the point. This called a derivative.

If you have a cube or a sphere, you can calculate the volume and surface area easily. If you have an odd shape, you need to use calculus. You use calculus to make a infinite number or really small slices of the object you’re interested in, determine to sizes of the slices, and then add all those sizes up. This process is called integration. It turns out that integration is the reverse of derivation (finding a derivative).

In summary, calculus is a tool that lets you do calculation with complicated curves shapes, etc. that you would normally not be able to do with just algebra and geometry.


Differentiation and Integration

Differentiation is the process of obtaining the derived function f′(x) from the function f(x), where f′(x) is the derivative of f at x.

The derivatives of certain common functions are given in the Table of derivatives,


Table of derivatives :
 

f(x) f'(x)
xn nxn-1
sin x cos x
cos x -sin x
tan x sec2 x
cot x -cosec2…

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