Geometrical differential calculus

The study of differential geometry involves the analysis of curvatures that can be approximated locally using straight lines. The study of differential calculus involves the analysis of the functions. Calculus functions are single-valued curves with single-valued branches in a coordinate system, where the horizontal variable is responsible for the vertical variable. However, students need to be aware of the significant differences between them. When it comes to differential geometry, vertical and horizontal units share the same dimensions. The vertical and horizontal units are typically different in differential calculus, e.g., height and time.

Some Basic Differences in differential Geometry vs. Calculus

• There are some differences in both studies about the distance between the points. In differential geometry, the Pythagorean Slant Distance formula is the most popular. There is no notion of the slant distance in the differential calculus plane.

• The derivative is a subject with a different significance in both subjects. 

• In differential geometry, The slope of the tangent lines will determine the line’s direction or the angle formed with Horizontal Axis.

•  In differential calculus, there is no notion of direction; instead, the derivative defines a change in a rate.

• In differential geometry calculus, the line described by the equation is y = x.

It subtends an angle 45deg with the angle horizontal. However, calculating the linear relationship of h = t is not a factor.

In most cases, the three words derivative, slope, and the rate of change are employed to describe the

Three important local characteristics of points on an axis in differential geometry:

• Position,
• direction,
• curvature or turning. 

In differential geometry, local curvature is a result of the angle that changes as the distance is increased along the curve.

Differential Geometry

In differential geometry, both horizontal and vertical variables such as x and y share the identical

dimensional units:

• In the graph of equations, y = 1 – x2 is the upper portion of a circle having a radius of 1.
• The equation’s graph y = x will appear straight with 45 degrees to the

horizontal and an angle of 1.

•  the direction of straight lines can be determined using either angle measured with horizontal or m or the slope. Note the formula y = y0: m(x – the number x0) and Dy = m Dx.
• The smallest shifts in the distance along the curve may be calculated using the Pythagorean slant.

Distance equation: ds = Dx2 + Dy2 .

• The curves considered could be considered geometric shapes.
• The angle of intersection between curves will be measured using the angle difference Between the lines of tangency at the intersection point.

Differential Calculus:

The differential equations in differential calculus. The variables in the describing equations represent the quantities that have units. They are generally different.

Definition of Derivatives:

The meaning of the geometrical y = f(x) is the slope of the tangent of that curve, namely y = f(x) at ( x, f(x)). The primary differentiation method involves calculating the derivative of the equation by using limits. Boundaries. Let a function on the curve include the equation y = f(x). Let’s consider an instance of a point P having coordinates(x, f(x)) on the curve. Another point, Q, is a point with the coordinates (x+h, f(x+h)) on the curve. Then PQ is the second of the curve. The slope of the curve at a specific place is the slope of the horizontal line that runs through the location. We know that the slope of the secant lines is y2-y1x2-x1.

What is the Differentiation Formula?

The derivatives of functions can be found with the derivative formula’s help, which was calculated in the preceding section. The derivatives of basic functions are regarded in the form of the differentiation formula.

Let’s consider a function y = xn , n > 0.

=> f(x + Δx) = (x + Δx)n , f(x + Δx)-f(x) = (x + Δx)n – xn

f(x+δx)−f(x)δx=limδx→0(x+δx)n−xn(x+δx)−x=limy→xyn−xny−x=nxn−1,

Here y = x + Δx and y → x as Δx → 0.

like this, we can derive the derivatives of the other exponential and trigonometric functions using this differentiation function

Differentiation of The 

Elementary Functions

• The derivative of a constant function is 0

=> if y = k, 

where k =constant, 

then y’ = 0.

• The derivative of a power function

=> If y = xn , n > 0. Then y’ = n x n-1

• The derivative of logarithmic functions

 =>If y = lnx, then y’ = 1/x and if y = logax, then y’ = 1/[(log a) x]

• The derivative of an exponential function

=>If y = a x , 

y = ax log a

Differentiation of The Trigonometric Functions

• when y = sin x=> y’ = cos x
• when y = cos x => y’ = -sin x
• when y = tan x => y’ = sec2 x
• when y = sec x =>y’ = sec x tan x
• when y = cosec x => y’ = -cosec x cot x
• when y = cot x => y’ = -cosec*2 x

Differentiation of Inverse Trigonometric Functions

• If y = sin-1 x, y’ = 1/√(1−x2)

• If y = cos-1 x, y’ = −1/√(1−x2)

• If y = tan-1 x, y’ = 1/(1+x2)

• if y = cot-1 x, y’ =−1/(1+x2)

• If y = sec-1 x, y’ = 1/x√(x2−1)

If y = cosec-1 x, y’ = −1/x√(x2−1)

Conclusion:

The subject of geometrical differential calculus evolved into an area studied as an independent field of study and distinct from the general idea of analytic geometry. This was in the 1800s. Differential geometry differentiation formulas have applications across sciences and maths. The most prominent application of this language was utilized in the work of Albert Einstein in his theory of general relativity. It was also used then by physicists to aid in creating quantum field theory and the most widely used particle physics model. Beyond physics, it has applications in chemical sciences, engineering, economics, control theory, computer graphics and vision, and, more recently, the field of machine learning.