# How does calculus relate to dimensions?

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paulo84

I am trying to understand what time^2 and velocity^2 mean in terms of how to visualize them? This wasn't explained in Physics or Mechanics (Further Mathematics) in high school, unfortunately. It seems likely it relates to matrices, maybe?

Appreciate any replies! :)

## Answers and Replies

PetSounds
For acceleration, you're better to think of it as metres per second per second rather than metres per second squared. For example: In freefall, your speed increases at a rate of 9.8 m/s per second. Mathematically, this is the same thing as 9.8 m/s2, but it's a little easier to wrap your head around.

paulo84
For acceleration, you're better to think of it as metres per second per second rather than metres per second squared. For example: In freefall, your speed increases at a rate of 9.8 m/s per second. Mathematically, this is the same thing as 9.8 m/s2, but it's a little easier to wrap your head around.

Thanks. It's just... a square is a shape? Or not necessarily?

Matthew314159271828
There isn't any way to "visualize" why there is a velocity squared term in the kinetic energy equation. Most equations in physics are in a mathematically simplified form that doesn't reveal the laws and principles that were used to derive them. Moral: Understanding how the equation is derived is more valuable than trying to "visualize" its individual terms.

russ_watters
PetSounds
Thanks. It's just... a square is a shape? Or not necessarily?
To square a number means to multiply a number by itself. It doesn't always refer to a geometric shape.

paulo84
To square a number means to multiply a number by itself. It doesn't always refer to a geometric shape.

But isn't the number we get from squaring only known because of that shape? Or not?

There isn't any way to "visualize" why there is a velocity squared term in the kinetic energy equation. Most equations in physics are in a mathematically simplified form that doesn't reveal the laws and principles that were used to derive them. Moral: Understanding how the equation is derived is more valuable than trying to "visualize" its individual terms.

OK...I'm visualizing hidden squares! Folded back on themselves...I appreciate your point about the important part to understand, but do you think that would be correct??

Mentor

I am trying to understand what time^2 and velocity^2 mean in terms of how to visualize them? )
I am assuming that you know the difference between vectors and scalars. Energy is a scalar and velocity is a vector, so if you want an energy which is related to velocity then you need some operation which takes a vector and returns a scalar.

That operation is known as the dot product. So ##v^2## is shorthand for ##v \cdot v = v_x^2 + v_y^2 + v_z^2##. This is the square of the speed.

As far as visualization, it means that if the x-axis is speed and the y-axis is energy then the graph is a parabola with the vertex at the origin

paulo84
paulo84
I am assuming that you know the difference between vectors and scalars. Energy is a scalar and velocity is a vector, so if you want an energy which is related to velocity then you need some operation which takes a vector and returns a scalar.

That operation is known as the dot product. So ##v^2## is shorthand for ##v \cdot v = v_x^2 + v_y^2 + v_z^2##. This is the square of the speed.

As far as visualization, it means that if the x-axis is speed and the y-axis is energy then the graph is a parabola with the vertex at the origin

Wow, thanks, that's awesome! Now I have some reading to do.

Matthew314159271828
Look, there is no way to visualize what the equation is conveying in a physical sense. The mathematical visualization in terms of a graph is not important to the physicist.

PetSounds
But isn't the number we get from squaring only known because of that shape? Or not?

Why? You can multiply 3 × 3 without involving geometry.

paulo84
Look, there is no way to visualize what the equation is conveying in a physical sense. The mathematical visualization in terms of a graph is not important to the physicist.

Oh sorry, I should have asked the question in the Maths subforum.

Mentor
The mathematical visualization in terms of a graph is not important to the physicist.
I wouldn’t go that far. If graphs and visualizations weren’t important to physicists then there wouldn’t be so many of them in physics papers and textbooks.

Matthew314159271828
You're absolutely right

paulo84
I am assuming that you know the difference between vectors and scalars. Energy is a scalar and velocity is a vector, so if you want an energy which is related to velocity then you need some operation which takes a vector and returns a scalar.

That operation is known as the dot product. So ##v^2## is shorthand for ##v \cdot v = v_x^2 + v_y^2 + v_z^2##. This is the square of the speed.

As far as visualization, it means that if the x-axis is speed and the y-axis is energy then the graph is a parabola with the vertex at the origin

Dale - I came back to your post. So the vector velocity is the sum of the speed along each of x, y and z axes?

Relative to acceleration, is there an implication that you're dealing with 2 dimensions of time, or not?

paulo84
Sorry, mixing up my concepts. The square of velocity is the sum of the square of the speed along each of x, y and z axes?

paulo84
And likewise velocity is essentially speed in 3 dimensions, which can be expressed as a matrix?

Jayalk97
And likewise velocity is essentially speed in 3 dimensions, which can be expressed as a matrix?
Look my dude, you're overcomplicating this in your head. Let's start from the bottom. Position is where you are. The rate at which you change position is velocity. The rate at which you change velocity is acceleration. Change in position is distance per second. We can denote that as meters per second, or meters/seconds. Acceleration as the rate at which your velocity changes. This could be denoted as velocity per second, or velocity/seconds. If you sub in the definition of meters per second into that you get meters/(second^2).

Jayalk97
Look my dude, you're overcomplicating this in your head. Let's start from the bottom. Position is where you are. The rate at which you change position is velocity. The rate at which you change velocity is acceleration. Change in position is distance per second. We can denote that as meters per second, or meters/seconds. Acceleration as the rate at which your velocity changes. This could be denoted as velocity per second, or velocity/seconds. If you sub in the definition of meters per second into that you get meters/(second^2).
I should also include that velocity has two components, speed and direction. Depending on the number of physical spatial dimension you are working with this, the direction part can have multiple components, one for every direction you're able to move in.

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paulo84
Look my dude, you're overcomplicating this in your head. Let's start from the bottom. Position is where you are. The rate at which you change position is velocity. The rate at which you change velocity is acceleration. Change in position is distance per second. We can denote that as meters per second, or meters/seconds. Acceleration as the rate at which your velocity changes. This could be denoted as velocity per second, or velocity/seconds. If you sub in the definition of meters per second into that you get meters/(second^2).

Thanks. I need to look at it in 3D to understand it! I was told in school I was a natural mathematician and a natural chemist but not a natural physicist. I only made it through the Mechanics paper because I could work out the inbetween maths, I sometimes have trouble grasping concepts which seem simple to others...

paulo84
I should also include that velocity gas two components, speed and direction. Depending on the number of physical spatial dimension you are working with this, the direction part can have multiple components, one for every direction you're able to move in.

Tensors?

Jayalk97
Tensors?
Not really quite sure what you mean by that but velocity is often represented as a vector, with a direction (where the arrow literally points) and the speed represented by magnitude (how long the arrow is). Sorry if I sound patronizing I don't really know where you are in your physics education so I want to spell everything out to make sure I'm clear.

paulo84
Not really quite sure what you mean by that but velocity is often represented as a vector, with a direction (where the arrow literally points) and the speed represented by magnitude (how long the arrow is). Sorry if I sound patronizing I don't really know where you are in your physics education so I want to spell everything out to make sure I'm clear.

You don't sound patronizing...https://www.physicsforums.com/threads/vectors-and-matrices.936357/

Jayalk97
Good lol. But as for matrices, everything relates to matrices. They're just a way to display information. If you could find a way to put some set of information into a matrix, you could then easily manipulate that information to achieve a desired result. Matrices and linear algebra as a whole are used everywhere. For instance as an electrical engineering student I use them to solve for current and voltages in certain circuits. By storing the information I get from a circuit into a matrix I could often literally plug it all into a calculator and get answer that might take me 10 minutes of algebra otherwise.

paulo84
Good lol. But as for matrices, everything relates to matrices. They're just a way to display information. If you could find a way to put some set of information into a matrix, you could then easily manipulate that information to achieve a desired result. Matrices and linear algebra as a whole are used everywhere. For instance as an electrical engineering student I use them to solve for current and voltages in certain circuits. By storing the information I get from a circuit into a matrix I could often literally plug it all into a calculator and get answer that might take me 10 minutes of algebra otherwise.

The problem is it gets really really complicated when you have to put in inverse values into tensors and you can't just use '1's but have to work out the square root of everything. Well, maybe it's not that complicated?

Jayalk97
The problem is it gets really really complicated when you have to put in inverse values into tensors and you can't just use '1's but have to work out the square root of everything. Well, maybe it's not that complicated?
Not my area of expertise haha. By the look of it you're taking some sort of solid mechanics class? If it doesn't require you to actively use linear algebra I'd just throw it all into a calculator and save time.

paulo84
Not my area of expertise haha. By the look of it you're taking some sort of solid mechanics class? If it doesn't require you to actively use linear algebra I'd just throw it all into a calculator and save time.

I meant cube roots. I'm not taking any class... :X

Mentor
So the vector velocity is the sum of the speed along each of x, y and z axes?
Are you familiar with vectors? This question looks like you have never used vectors.

If not, then this is absolutely priority number 1! Vectors are necessary for physics at all levels

Relative to acceleration, is there an implication that you're dealing with 2 dimensions of time, or not?
Definitely not

paulo84
Are you familiar with vectors? This question looks like you have never used vectors.

If not, then this is absolutely priority number 1! Vectors are necessary for physics at all levels

Definitely not

I'm 33 and haven't dealt with vectors in maths since I was 17, unfortunately. Until very recently, that is...

Mentor
My recommendation is to go back and review both vectors and basic differential and integral calculus. You can’t do any physics without vectors, you can do a little with just vectors and algebra, but most of what you are asking requires vectors and calculus too.

paulo84
Are you familiar with vectors? This question looks like you have never used vectors.

If not, then this is absolutely priority number 1! Vectors are necessary for physics at all levels

Definitely not

Can you tell me why 'definitely not'?

Jayalk97
Can you tell me why 'definitely not'?
It's just that you're thinking of units the wrong way. Honestly I would look at unit conversion lessons online usually in chemistry courses. Units don't need any sort of dimension attached to them.

Science Advisor
Homework Helper
It's just that you're thinking of units the wrong way. Honestly I would look at unit conversion lessons online usually in chemistry courses. Units don't need any sort of dimension attached to them.
It is conceivable that @paulo84 is not aware of the distinction between "dimension" as in dimensional analysis and "dimension" as the number of items required in a basis for a vector space.

Jayalk97
I feel like you were taught or taught yourself certain things out of order @paulo84 . I would take a book out and linear algebra and read through it. It's very straightforward and interesting and it should also give you some clarification on how matrices work and, if it's a good textbook, how they are used in practice.

Mentor
Can you tell me why 'definitely not'?
I know that you claim to have trouble distinguishing clocks and rulers, but one of the defining things about rulers is that you can orient up to three of them orthogonal to each other and measure distances in orthogonal directions. There are not two orthogonal directions of time that you can measure with clocks.

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Mentor
I feel like you were taught or taught yourself certain things out of order
Yes, he is going out of order despite repeated suggestions by multiple people that it will not be an effective approach.