Relationship of Geometry, Algebra, and Calculus: Assignment

This assignment explores the relationship between geometry and algebra within the context of calculus. It begins by defining the basic concepts of geometry, including points, lines, and dimensions, and then introduces algebra as a representation tool for geometric elements. The solution highlights how geometry, including one, two, and three-dimensional figures, relates to calculus through the use of algebraic representations of areas and volumes. The assignment further examines the core concepts of calculus, such as differentiation and integration, and how these relate to the changes in dimensions. It explains how calculus can be applied to calculate areas and rates of change, and it references the fundamental equation of directional derivatives to illustrate the connection between geometry, algebra, and calculus in calculating smaller changes. The assignment uses references to support the key concepts of the relationship.

1Running head: GEOMETRY AND ALGEBRA IN CALCULUS
Geometry and Algebra in Calculus
Name
Institutional Affiliation

2GEOMETRY AND ALGEBRA IN CALCULUS
Introduction
Geometry is a mathematical branch that deals with properties and how they are related to lines,
surfaces, and solids. Mostly geometry deals with measurements and how points, angles or lines
relate to one another (Mirriam-Webster, 2019). Algebra is more of a representation, where letters
are used to represent or symbolizes numbers (Russell, 2018). We want to see how algebra is
brought about by geometry and how the two are used in calculus. In geometry, starts from one
dimensional figure, which have only dots, these are not usually measurable. From dots, to
another one-dimensional figures called lines, which is one point connected to the next point, or
dot, it has no thickness and extends infinitely in both directions, whether perpendicularly,
vertically, or diagonally.
How Geometry and Algebra Relates to Calculus
Geometry deals with lines, which is one dimensional, areas, which are two dimensional, and
volumes which are three dimensional. In one dimensional diagram, there’s only the length aspect
of it, two dimensional there’s the length and the width, in three dimensions, we have the area and
now the third dimension which is the thickness or height, this gives us the volume. Geometry
gives us the dimensions of all these.
To find the relationship of the areas, volume and the lengths of objects, they can be expressed
algebraically, like the length of an item can be l, the width can be expressed as w, and the height
expressed as h. These when put together gives representation of areas and volume
Areaof a rectangle=l X w
The example above shows that to find an area you have to consider the length and width, which
are the two dimensions.
Areaof a ˚¿ π r2
On this the representation is the radius of the circle and a constant π.
One thing we see from all of these is how the algebra relates to geometry.
So, if we find how geometry relates to calculus, then we shall have met what our research
proposes.
Calculus
Mostly this studies how things change, where there are instantaneous rates, and accumulation of
quantities in other words, differentiation and integration respectively. These are the two major
branches of calculus and they are connected by the basic theories of calculus. Calculus can be
applied in the calculation of an area. Especially when the regular shapes are plotted on a
cartesian plane or the irregular shapes.
When looking at the rate of change of dimensions, then we are talking about how the length
changes and the width. Yet on the other hand, when we talk about the geometric algebra in
relation to calculus, we are drawn to the basic equation of the differentiation;

3GEOMETRY AND ALGEBRA IN CALCULUS
∇b F =lim
B → 0
m=lim
h→ 0
f ( x +h ) −f ( x )
h
Where F(a) is a multi-vector while h and x are vectors, therefore the above equation defines the
directional derivative of F(a) along h.
When geometry studies about shape, and algebra studies about how arithmetic operations relates
with different dimensions, these two are combined in the calculations of smaller changes, called
calculus.

4GEOMETRY AND ALGEBRA IN CALCULUS
References
Kline, M. (1956). The straight line. Sci,Amer.
Mirriam-Webster. (2019, Febriuary 19). Geometry. Retrieved from Meriam Webster:
https://www.merriam-webster.com/dictionary/geometry#synonyms
Russell, D. (2018, September 16). Algebra definition . Retrieved from Thought Co.:
https://www.thoughtco.com/definition-of-algebra-2311577