*Carpe Diem (Seize The Day)*: What is the relationship between Calculus and Physics?

How is the topic of physics and calculus related, and how do these topics depend on each other. For example, acceleration is taught in calculus even though it is a pure physics problem. What are some other instances in which these topics depend on each other? what topics? How exactly are they related?

I’ve noticed over several years, that even excellent math students find calculus and physics difficult, why do many find these topics difficult? Is it the mathematics or concepts that are hard to understand?

I thank all in advance. Thanks.

**Answers and Views:**

*Answer by $ $ $ $ *

Calculus is a very vague thing and damn confusing.

Very few people understand it.

Calculus mainly deals with rates of changing and physics has numerous stuff involving rates of change.

Eg: velocity.

*Answer by Don’t fear the reaper*

Calculus is very different from other math topics, it requires more thinking abstractly instead of memorization. the concepts are much harder to grasp.

Calculus and physics have been related from the beginning, Isaac Newton invented calculus specifically to solve physics problems about gravity and how to predict where and when objects move when they fall.

*Answer by China Jon*

Wow. You went to town on this relationship didn’t you? I broke your questions down into 5 questions:

1. How are the topics of physics and calculus related?

Physics is the science of matter and energy and of interactions between the two, grouped in traditional fields such as acoustics, optics, mechanics, thermodynamics, and electromagnetism, as well as in modern extensions including atomic and nuclear physics, cryogenics, solid-state physics, particle physics, and plasma physics.

Calculus is

1. The branch of mathematics that deals with limits and the differentiation and integration of functions of one or more variables.

2. A method of analysis or calculation using a special symbolic notation.

3. The combined mathematics of differential calculus and integral calculus.

So, Calculus is a tool that is used to answer questions in which there are one or more variable that can change over time. It was invented, mostly by Newton, to determine where planets would be on a future date, to verify theories of gravitation. Physics is a science which focuses on energy and time. Related energetic properties can change over time. It takes a very flexible and powerful tool to clearly and accurately find given answers at given times. We know that the the pull of gravity on Earth will accelerate an object to a speed of 32 feet per second after 1 second. But how fast is it going after 1/2 second, or 4.2 seconds?

2. How do these topics depend on each other?

The understanding of Physics depends on many tools including atom smashers, telescopes, computers, and mathematics. Without something to look at, a telescope is useless. Without problems of complexity, calculus would be as useless.

3. What are some topics or instances in which these topics depend on each other?

Acceleration is dependent on force and mass. A rocket engine provides force, but the mass goes down as the fuel is used up, so acceleration increases as the rocket goes up. How fast is it going after 25 seconds? How much mass does it have? How fast will it be going after 1 minute? How much fuel will be needed to accelerate a 1000 kg payload to 7,600 meters per second?

4. How exactly are they related?

Mass, speed, temperature, and gravity etc. are physical properties. How they relate is a mathematical relationship which can be solved with Calculus.

5. Why do many excellent math students find these topics difficult?

The Calculus is like three dimensional math. The relationships in Physics are often beyond our experience and counter intuitive. Good students are often very intuitive, an ability that lets them down when they try to understand Physics and the tools by which we study it.

6. Is it the mathematics or concepts that are hard to understand?

Yes.

;-D Get a good night’s sleep before studying either Physics of Calculus!

*Answer by darrenfoong1*

Relationship? I guess that many relationships in physics involved changes, and rate of changes lead us to calculus.

I believe calculus is the study of changes, and it has extended to such a big extent that it is not merely finding the derivative of a function.

I sort of think that calculus is the basis of physics, but that’s me. Various relationships encountered in physics involve a simple differential operator. And it explains the laws in a nice way.

Say, velocity is the rate of change of displacement with time. We can write it so easily with v = ds/dt. There’s probably no other concise way to express velocity.

Mathematics is a subject by itself, but to fully understand physics, one must have a good grasp of the underlying mathematics and must be able to understand the concept.

For example, you might be excellent in maths, but if you don’t get even a part of general relativity (or can’t imagine it), then it’s not gonna look good.

Just some thoughts.

*Answer by Lubricated Shafer*

s = x

v = d (s) / dt

a = d (v) / dt

v = rate of change of position, with respect to time

accel = rate of change of vel., with respect to time

These are concepts which are easy to understand.

It’s applying mathematical formulation to physical scenarios in the real world that i find difficult. Hence your question I suppose

*Answer by Swapan G*

ANSWER TO ALL THOSE ASKED AND NOT ASKED QUESTIONS as FOLLOWS :

PHYSICS , LIKE ALL OTHER SCIENCE SUBJECT, IS BASED ON REASONING OF CAUSE AND EFFECT OF NATURAL PHENOMENON . THE REASONING IS BEST EXPLAINED BY MATHEMATICAL PRINCIPLES . MATHEMATICS AGAIN IS BASED ON PHILOSOPHICAL TRUTHS FORMULATED IN NUMBERS.

SOME NUMBERS ARE DESCRETE AND WE USE THEM IN OUR DAY TO DAY LIFE AND EASILY UNDERSTOOD . TRANSITION OF ONE NUMER TO ANOTHER IS EXPLAINED BY DECIMALS. FOR FINER TRANSITION WE NEED SOME CONTINUOS FUNCTION WHICH MAY REPRESENT CONTINUITY OF NATURAL PHENOMENON AT ANY TIME INSTANT . CALCULUS DEALS WITH SUCH FUNCTIONS OF VARIABLES ANALYTICALLY .

SOME PERSONS ARE GOOD WITH REAL DIGITAL MATHEMATICS BUT NOT SO COMFORTABLE WITH ANALYTICAL ONE JUST LIKE SOME ARE GOOD AS SCIENTISTS , SOME ARE BETTER AS BUSSINESSMAN.

Those with analytical bend of mind will be more comfortable with calculus and physics among intelligent student community.

*Answer by qspeechc*

It just so happens in our universe that the universe can be explained and understood in terms of mathematics. Why this should be the case, nobody knows- that is a meta-physical question. Perhaps we humans need something like mathematics to help us understand the universe, we have used mathematics to analyse, axplain and understand the universe. Or maybe there is a more fundamental connection. Maybe “Mother Nature” is a mathematician…

Acceleration doesn’t need to be taught in calculus. A calculus course could be completely abstract, without reference to the real world. But, calculus has found an extremely wide range of application, so it makes sense to bring in the applications in a calculus course, not least because those taking the course may want to apply to real world problems (physicists, engineers, even business men), but it also helps heuristically if it relates to the real world, things that are tangible and ‘knowable’.

That is to say, calculus does not depend on physics. Calculus to be completely abstract. But it is quite unlikely that physics would have gotten far without calculus. Just about every branch of physics can be dealt with within the framework of calculus- dynamics, kinematics, hydraulics…- you name it. Some topics in physics would even be impossible without calculus, such as variable acceleration. Even in quantum mechanics, where (almost) everything is discrete, calculus plays an important role.

But, one could say that certain physics problems which required an analysis with calculus, sort of spurred on the developement of calculus, in a similar way that engineering problems pushed physics forward (think Fourier Series etc.).

Why does calculus find so much application in physics? Calculus basically deals with infinitesimal changes- changes that are not zero, but smaller than any imaginable real number. In physics (reality) the universe operates with infinitesimal changes. So calculus (specifically, infinitesimal calculus) works splenidly with the real world where things can be analysed infinitesimally. And such infinitesimal analyses covers the (usual) situation where the quantity in question is not constant, or not even changing at a constant rate, which would not be possible without calculus.

And that would explain why calculus and physics are difficult. Calculus requires thinking about infinitesimal changes, which sound quite contradictory and mind-boggling. Calculus is unlike any other branch in mathematics. Also, physics requires a sort of “visualisation” and intuition about physical reality.

One must also remember that the topics covered in one physics or calculus course (at university) may have taken thinkers and scientists centuries, and even millenia, to come to grips with, and all that thought is condensed into a semester or one year.

It makes you think….

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