How does calculus relate to dimensions?

[paulo84]

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.

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.

 

[paulo84]

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.

 

[jbriggs444]

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.

 

[paulo84]

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.

 

[jbriggs444]

I have more reading to do.

Relevant content is at https://en.wikipedia.org/wiki/Vector_space#Basis_and_dimension.

Despite seeming nothing at all like the high school physics concept of vectors, the mathematical formalism is pretty cool and has high school vectors as a subset.

 

[David Lewis]

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).

 

[votingmachine]

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.

 

[Mark44]

The OP is no longer with us, so as his question as been answered, I'm closing this thread.