How does calculus relate to dimensions?

In summary: I appreciate your point about the important part to understand, but do you think that would be correct??I am trying to understand what time^2 and velocity^2 mean in terms of how to visualize them? )Appreciate any replies! :)In summary, time^2 and velocity^2 are related to how to visualize them scalar-wise. If you want an energy which is related to velocity, you need to use the dot product which is the square of the speed.
  • #36
jbriggs444 said:
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.

Indeed.

Dale said:
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.

That makes sense. I am beginning to do more reading, starting with maths before physics.
 
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  • #37
jbriggs444 said:
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.

I was thinking about this and hope I understand a little better. One value only defines 0 dimensions - an object in no space and time. 2 values define 1 dimension - an object and a length. 3 values define 2 dimensions - an object, a length, and a direction. You need 4 values to define 3 dimensions.
 
  • #38
paulo84 said:
I was thinking about this and hope I understand a little better. One value only defines 0 dimensions - an object in no space and time. 2 values define 1 dimension - an object and a length. 3 values define 2 dimensions - an object, a length, and a direction. You need 4 values to define 3 dimensions.
One value is one dimension. For instance, position on a narrow road.
Two values is two dimensions. For instance, position on a flat plane.
Three values is three dimensions. For instance, position and altitude.
Four values is four dimensions. For instance, position, altitude and time.

The notion is nailed down in the field of "linear algebra" where one learns a formal definition for vectors, scalars, vector spaces and basis vectors. The dimension of a vector space is the number of vectors that must appear in a basis for that space. Or, equivalently, the number of coordinates required to specify an arbitrary point using that basis.

In the trivial vector space consisting of a single point represented by the zero vector, it takes no coordinates at all to specify the only point there is. The dimension of the space is zero.
 
  • #39
jbriggs444 said:
One value is one dimension. For instance, position on a narrow road.
Two values is two dimensions. For instance, position on a flat plane.
Three values is three dimensions. For instance, position and altitude.
Four values is four dimensions. For instance, position, altitude and time.

The notion is nailed down in the field of "linear algebra" where one learns a formal definition for vectors, scalars, vector spaces and basis vectors. The dimension of a vector space is the number of vectors that must appear in a basis for that space. Or, equivalently, the number of coordinates required to specify an arbitrary point using that basis.

In the trivial vector space consisting of a single point represented by the zero vector, it takes no coordinates at all to specify the only point there is. The dimension of the space is zero.

I have more reading to do.
 
  • #40
  • #41
PetSounds said:
To square a number means to multiply a number by itself. It doesn't always refer to a geometric shape
The term square in this context is used figuratively (the area of a square varies with the length of its side the same as x2 varies with x).
 
  • #42
Unit analysis is important, but the units are not true "dimensions".

Say I have 10 chickens and they lay 10 eggs each day. I can say:
1 chicken = 1 egg/day
I'm getting 10 eggs/day from 10 chickens.

Say I start eating one chicken per day.
My egg yield goes down 1 egg/day EACH DAY.
I see a reduction of 1 egg per day per day ... 1 egg/day^2

You can create equations that the units get weirder and weirder. I could introduce chicken re-population. And predators. At some point I might have units of wolves/egg^3 ... it does not mean that there is a cubic egg thing in the world ... just the units in the equations have that many layers.

The geometric interpretation of the units is not right. The units are truly best treated in the math as "squared" but they are not geometric squares.

EDIT: It is confusing. Sometimes it can be very productive to imagine units as dimensions. If I tell you a pressure is 1 PSI, you can very productively imagine a square with sides 1-inch, and a pound weigh placed on it. If I tell you water has density of 1 g/cm^3, you can productively imagine a 3 dimensional cube with sides of a centimeter, and a scale showing 1 gram.

With acceleration, m/s^2, it is not productive to imagine an area representation of time. We don't think that way. The math works to let us square the time, and get the distance. But I don't think of it as a time-area relationship.
 
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  • #43
The OP is no longer with us, so as his question as been answered, I'm closing this thread.
 
<h2>1. How does calculus relate to dimensions?</h2><p>Calculus is a branch of mathematics that deals with the study of change and motion. It is commonly used to analyze and solve problems involving variables, rates of change, and areas or volumes. Since dimensions describe the size, shape, and position of an object, calculus is used to understand and manipulate these properties in various dimensions.</p><h2>2. How does calculus help in understanding higher dimensions?</h2><p>Calculus provides a framework for understanding and working with higher dimensions, which are beyond the three dimensions we experience in our daily lives. It allows us to visualize and analyze complex shapes and spaces in multiple dimensions, such as 4D or 5D objects, by using mathematical concepts like derivatives, integrals, and limits.</p><h2>3. Can calculus be used to solve problems in multiple dimensions?</h2><p>Yes, calculus can be applied to solve problems in multiple dimensions. It provides tools for finding maximum and minimum values, optimizing functions, and calculating areas and volumes in higher dimensions. It is also used in fields like physics, engineering, and economics to model and solve real-world problems in multiple dimensions.</p><h2>4. How does the concept of limits relate to dimensions in calculus?</h2><p>Limits are an essential concept in calculus that helps us understand the behavior of a function as the input or output values approach a certain value. In higher dimensions, limits are used to describe the behavior of a function as it approaches a point in a multi-dimensional space. This allows us to analyze and solve problems involving rates of change and optimization in multiple dimensions.</p><h2>5. Can calculus be used to understand the concept of infinity in higher dimensions?</h2><p>Yes, calculus is used to understand and work with the concept of infinity in higher dimensions. The concept of limits is crucial in this regard, as it helps us understand how a function behaves as the input or output values approach infinity. This is important in fields like geometry and physics, where infinity is often used to describe infinite shapes and spaces in higher dimensions.</p>

1. How does calculus relate to dimensions?

Calculus is a branch of mathematics that deals with the study of change and motion. It is commonly used to analyze and solve problems involving variables, rates of change, and areas or volumes. Since dimensions describe the size, shape, and position of an object, calculus is used to understand and manipulate these properties in various dimensions.

2. How does calculus help in understanding higher dimensions?

Calculus provides a framework for understanding and working with higher dimensions, which are beyond the three dimensions we experience in our daily lives. It allows us to visualize and analyze complex shapes and spaces in multiple dimensions, such as 4D or 5D objects, by using mathematical concepts like derivatives, integrals, and limits.

3. Can calculus be used to solve problems in multiple dimensions?

Yes, calculus can be applied to solve problems in multiple dimensions. It provides tools for finding maximum and minimum values, optimizing functions, and calculating areas and volumes in higher dimensions. It is also used in fields like physics, engineering, and economics to model and solve real-world problems in multiple dimensions.

4. How does the concept of limits relate to dimensions in calculus?

Limits are an essential concept in calculus that helps us understand the behavior of a function as the input or output values approach a certain value. In higher dimensions, limits are used to describe the behavior of a function as it approaches a point in a multi-dimensional space. This allows us to analyze and solve problems involving rates of change and optimization in multiple dimensions.

5. Can calculus be used to understand the concept of infinity in higher dimensions?

Yes, calculus is used to understand and work with the concept of infinity in higher dimensions. The concept of limits is crucial in this regard, as it helps us understand how a function behaves as the input or output values approach infinity. This is important in fields like geometry and physics, where infinity is often used to describe infinite shapes and spaces in higher dimensions.

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