Tuesday, October 18, 2011

∑ F = ma

… is a differential equation:

where acceleration a(t), velocity v(t), and displacement s(t) are all vectors and functions of time. This equation is second-order in position because the highest derivative is the second time derivative of position. Combined with the right boundary conditions, s(t) (also called the trajectory: path through space and time) can be determined.

This differential equation can be solved one component, or dimension, at a time. Let us focus on one of these, and call it the x component. The equations for y and z can be found exactly the same way.

Constant acceleration

If the graph of a(t) signifying acceleration in the x direction is constant

then the graph of v(t), the velocity in the x direction, is a straight line with slope a0

and the graph of x(t), the position along the x axis, is a parabola

It is also possible for the acceleration, or either of the initial velocity or initial position, to be negative. Thus the displacement/projectile motion formula is derived.

Friday, August 12, 2011
After being introduced to the concept of a limit and the derivative, the typical calculus student is asked to evaluate simple antiderivatives and apply a strange symbol, ∫, much like an elongated “S”, to their notation. He then completely shifts gears, and applies his knowledge of limits to summations of rectangular areas, of which there are an infinite amount, and all with vanishingly small width. After establishing the techniques used in finding exact areas bounded by curves, he is asked again to apply ∫ to the function whose area it is he must calculate…and is left to his own devices to interpret the Fundamental Theorem of Calculus (FTOC) — the theorem which relates the derivative to bounds of a definite integral, and the area bounded by a function to its antiderivative. Few introductory calculus courses take the time to prove the theorem, and simultaneously probe the intricate connections that definite integration has with differential calculus.
The first part of the FTOC is as follows: a function

is a general antiderivative of f such that A’(x) = f(x). The proof of this part is lengthy, but less conceptually rigorous than the second.
Second part of FTOC — Prove:

Proof: Consider F, a function which is a general antiderivative of f. Also consider an antiderivative of f, A — for simplicity, the same as that which was used in the first part of the FTOC. Because F and A are both antiderivatives of f which differ by a constant of integration, C, you may then make a relation such that

where C is a constant. Substituting x = a into this equation brings you to the following relation:

(by early properties of definite integrals, the region has a width of 0). Substituting x = b,

which, when rearranged, yields

(Note: there are many proofs of the FTOC and its corollaries, so it’d be wise to browse through as many as you can find!)
It helps to introduce a bit of generalization for the purpose of understanding the geometric significance of the FTOC (coupled with the picture above): the gradient of a small target region of a function is approximated by ∆y/∆x, whereas the area of that small region may be approximated by ∆y•∆x — inverse operations, correlating directly to the inverse relationship between differentiation and integration.

After being introduced to the concept of a limit and the derivative, the typical calculus student is asked to evaluate simple antiderivatives and apply a strange symbol, ∫, much like an elongated “S”, to their notation. He then completely shifts gears, and applies his knowledge of limits to summations of rectangular areas, of which there are an infinite amount, and all with vanishingly small width. After establishing the techniques used in finding exact areas bounded by curves, he is asked again to apply ∫ to the function whose area it is he must calculate…and is left to his own devices to interpret the Fundamental Theorem of Calculus (FTOC) — the theorem which relates the derivative to bounds of a definite integral, and the area bounded by a function to its antiderivative. Few introductory calculus courses take the time to prove the theorem, and simultaneously probe the intricate connections that definite integration has with differential calculus.

The first part of the FTOC is as follows: a function

is a general antiderivative of f such that A’(x) = f(x). The proof of this part is lengthy, but less conceptually rigorous than the second.

Second part of FTOC — Prove:

Proof: Consider F, a function which is a general antiderivative of f. Also consider an antiderivative of f, A — for simplicity, the same as that which was used in the first part of the FTOC. Because F and A are both antiderivatives of f which differ by a constant of integration, C, you may then make a relation such that

where C is a constant. Substituting x = a into this equation brings you to the following relation:

(by early properties of definite integrals, the region has a width of 0). Substituting x = b,

which, when rearranged, yields

(Note: there are many proofs of the FTOC and its corollaries, so it’d be wise to browse through as many as you can find!)

It helps to introduce a bit of generalization for the purpose of understanding the geometric significance of the FTOC (coupled with the picture above): the gradient of a small target region of a function is approximated by ∆y/∆x, whereas the area of that small region may be approximated by ∆y•∆x — inverse operations, correlating directly to the inverse relationship between differentiation and integration.

Thursday, July 21, 2011
(Continued from previous post on the del operator, here)
When ∇ is applied to a vector-valued function, it takes the form of what is called the divergence of the vector function, V(x,y,z), when V is defined by V(x,y,z) = P(x,y,z) i + Q(x,y,z) j + R(x,y,z) k, where P, Q, and R, are scalar functions. The divergence is the dot-product of the del operator and V:

The divergence of V may consequently be thought of   as a scalar field. The physical significance of the divergence of a   vector field is that it is the amount of density “flowing” into, or out   of, a given region of space (visualizing the vector field as a quasi   fluid-like entity — consequences in electromagnetism and fluid dynamics   follow immediately). Alternatively, it may be visualized as how a   particular point behaves as a “source” from a point (in which density is   decreasing) or a “sink” toward a point (in which density is   increasing).
Another way in which ∇ is applied to vector-valued functions is the scenario in which, instead of the dot-product, the cross-product (or vector-product) of ∇ and the vector function V is taken — creating what is known as the curl of V:

(Note: the matrix determinant does not directly correlate to the   definition of the operation, but it is merely a method of better   remembering how to carry out the curl calculation.) The curl represents   the infinitesimal rotation of a 3-dimensional vector field. This   rotation is represented by a vector, whose length is proportional to the   magnitude of rotation, and whose direction is that of said rotation   about V. The intuitive interpretation of this operation   may be represented as this: if, in a vector field describing the  motion  of a viscous fluid; there is a ball with an imperfect surface,  then the  ball will rotate due to the motion of the fluid about it.
The Laplace operator (also known as the Laplacian) is a differential operator which is   defined to be the divergence of the gradient of a scalar field (in   Euclidean space):

The Laplace operator has analogs in many coordinate systems, and for   that reason its intuitive interpretations have many forms. In physical   systems, the Laplace operator may be interpreted as the flux density (flux per unit area) of the gradient flow of a scalar field. Common   applications lie in the analysis of gravitational potential fields,   electric potential fields, and heat flow. This operator gives rise to   the d’Alembert operator, which is its analog in 4-dimensional space-time. albanhouse delves into this particular operator’s connection to electromagnetism and tensor analysis in this recent post on Maxwell’s equations.

(Continued from previous post on the del operator, here)

When ∇ is applied to a vector-valued function, it takes the form of what is called the divergence of the vector function, V(x,y,z), when V is defined by V(x,y,z) = P(x,y,z) i + Q(x,y,z) j + R(x,y,z) k, where P, Q, and R, are scalar functions. The divergence is the dot-product of the del operator and V:

The divergence of V may consequently be thought of as a scalar field. The physical significance of the divergence of a vector field is that it is the amount of density “flowing” into, or out of, a given region of space (visualizing the vector field as a quasi fluid-like entity — consequences in electromagnetism and fluid dynamics follow immediately). Alternatively, it may be visualized as how a particular point behaves as a “source” from a point (in which density is decreasing) or a “sink” toward a point (in which density is increasing).

Another way in which ∇ is applied to vector-valued functions is the scenario in which, instead of the dot-product, the cross-product (or vector-product) of ∇ and the vector function V is taken — creating what is known as the curl of V:

(Note: the matrix determinant does not directly correlate to the definition of the operation, but it is merely a method of better remembering how to carry out the curl calculation.) The curl represents the infinitesimal rotation of a 3-dimensional vector field. This rotation is represented by a vector, whose length is proportional to the magnitude of rotation, and whose direction is that of said rotation about V. The intuitive interpretation of this operation may be represented as this: if, in a vector field describing the motion of a viscous fluid; there is a ball with an imperfect surface, then the ball will rotate due to the motion of the fluid about it.

The Laplace operator (also known as the Laplacian) is a differential operator which is defined to be the divergence of the gradient of a scalar field (in Euclidean space):

The Laplace operator has analogs in many coordinate systems, and for that reason its intuitive interpretations have many forms. In physical systems, the Laplace operator may be interpreted as the flux density (flux per unit area) of the gradient flow of a scalar field. Common applications lie in the analysis of gravitational potential fields, electric potential fields, and heat flow. This operator gives rise to the d’Alembert operator, which is its analog in 4-dimensional space-time. albanhouse delves into this particular operator’s connection to electromagnetism and tensor analysis in this recent post on Maxwell’s equations.

Tuesday, July 19, 2011
The del operator, denoted with what is called the nabla symbol (an inverted delta), is a differential operator connecting differential calculus of functions to the study of vectors, and vector-valued functions. The del operator has several forms, and is defined by

where ∂/∂x indicates the partial derivative with respect to x (similarly for y and z), and i, j, and k indicate the three standard unit vectors (all in the Cartesian coordinate system).
To a scalar function, F(x,y,z), in the Cartesian coordinate system, the del operator may be applied to create what is known as the gradient of F — defined as

The inclusion of the unit vectors in ∇ lead to the gradient’s being a vector-valued function in three dimensions, and the vector consequently is directed toward the greatest increase in F, at any point (x,y,z). Its magnitude is equal to the maximum rate of increase, and hence may be used as an analog to the 1-dimensional derivative, in 3 dimensions.
When ∇ is applied to a scalar field (function) of the form F(x,y,z), and a vector field a (= <a_1, a_2, a_3>) is chosen, the directional derivative takes the form

in which the dot-product of the gradient of F and and the vector a is taken. The most common analogy is this: if ∂F/∂x gives the rate of change of F in the x direction, then ∇F • a gives the rate of change of F, in vector form, toward the vector a. a is taken to be the unit vector for this calculation. This operation allows the rate of change of a scalar field with respect to an arbitrary — and sometimes changing — direction of a vector (a need not be a vector composed only of constant components), to be calculated. Its most common application for this operation lies in the field of fluid dynamics.
Coming soon: ∇applied to vector functions!

The del operator, denoted with what is called the nabla symbol (an inverted delta), is a differential operator connecting differential calculus of functions to the study of vectors, and vector-valued functions. The del operator has several forms, and is defined by

where ∂/∂x indicates the partial derivative with respect to x (similarly for y and z), and i, j, and k indicate the three standard unit vectors (all in the Cartesian coordinate system).

To a scalar function, F(x,y,z), in the Cartesian coordinate system, the del operator may be applied to create what is known as the gradient of F — defined as

The inclusion of the unit vectors in ∇ lead to the gradient’s being a vector-valued function in three dimensions, and the vector consequently is directed toward the greatest increase in F, at any point (x,y,z). Its magnitude is equal to the maximum rate of increase, and hence may be used as an analog to the 1-dimensional derivative, in 3 dimensions.

When ∇ is applied to a scalar field (function) of the form F(x,y,z), and a vector field a (= <a_1, a_2, a_3>) is chosen, the directional derivative takes the form

in which the dot-product of the gradient of F and and the vector a is taken. The most common analogy is this: if ∂F/∂x gives the rate of change of F in the x direction, then ∇Fa gives the rate of change of F, in vector form, toward the vector a. a is taken to be the unit vector for this calculation. This operation allows the rate of change of a scalar field with respect to an arbitrary — and sometimes changing — direction of a vector (a need not be a vector composed only of constant components), to be calculated. Its most common application for this operation lies in the field of fluid dynamics.

Coming soon: ∇applied to vector functions!

Tuesday, July 12, 2011
Many people have heard of Maxwell’s famous four equations:
∇·E = 0 ∇·B = 0 ∇ x E = -∂B/∂t ∇ x B = μ0J + μ0ε0(∂E/∂t)
But did you know that they can actually all be captured in one simple expression? When we extend classical electrodynamics into Minkowski 4-space (the geometry of special relativity), the time derivatives that appear in the derivation of Maxwell’s equations can be mixed into the same mathematics that takes care of the spatial expressions. By generalizing the Laplacian (∇2) to 4-space, we obtain the d’Alembertian (□2, sometimes written without the exponent). This operator allows us to rewrite Maxwell’s equations in tensor notation as one simple formula, sometimes called the Riemann–Sommerfeld equation:
□2Aμ = -μ0Jμ
Where Aμ is the four-potential (φ, cA) (with φ and A the familiar scalar and vector potentials) and Jμ the four-current (ρc, J) (with ρ and J the familiar charge and current densities), thereby using the power of differential geometry and tensor notation to capture all of classical electrodynamics in one swift stroke.
Background
Maxwell’s Equations describe how electric and magnetic fields relate to each other and how they relate to the presence of charged particles and currents. The way we’ve presented them above is in differential form, which describes how electric fields (E) and magnetic fields (B) change in space and time; they can also be presented in integral form, which describes how the total measure of these fields scales with quantities like current density (J).
Although these equations were originally formulated in terms of fields, physicists would later find that the equations could be much cleaner when expressed in terms of potentials. A similar revolution happened in classical mechanics; although Newton’s forces were a good start, Lagrangian dynamics showed us that understanding the potential energy of the system made solving for equations of motion much simpler than trying to add up dozens of force vectors. In electrodynamics, we have two kinds of potentials: the scalar potential φ (or, sometimes, V), which most people know by the term “voltage”; and the vector potential, A, which is related to magnetic fields and may be less familiar.
Tensors notation (for our purposes) is little more than a convenient way to represent vectors and matrices. When you read the Riemann-Sommerfeld equation, then, you’re really reading 4 equations at once. What are those equations? Repeat the equation four times but each time replace μ with a different value; so, one equation might be □2A1 = -μ0J1. The 1’s in this equation aren’t exponents, but rather tensor indices; just like you might see vector components written as vx, vy, and vz representing different components of v, A0, A1, A2, and A3 represent the four components of the (vector) tensor A.
What are the funny upside-down triangles? The symbol is called a nabla, and is often read as “del”. In some ways, it’s an abuse of notation, but it gives us a convenient way to write the gradient, divergence, curl, and Laplacian operators. The exercise for how this notation works is left to the reader, but note the definitions: ∇U = grad(U), the gradient; ∇·U = div(U), the divergence; ∇ x U = curl(U), the curl; and ∇2U = ∇·(∇U) = div(grad(U)), the Laplacian. Each of these functions has a very important place in vector analysis. Since we represent electric and magnetic fields mathematically as vector fields, these operations give us information about the geometry of those fields. The gradient function tells us the direction that a field changes most at each point; the divergence tells us how much the field is dispersing (a field where everything pointed away from one center point would have a high divergence); the curl tells us how much a field is swirling, so to speak; and the Laplacian tells us how much a field changes in strength as it disperses.
And finally, we have our partial derivatives (∂B/∂t, for example) that give us rates of change (like how a magnetic field B changes with respect to the time t), and our constants, ε0 and μ0. These constants have the interesting relation that, for the speed of light c, ε0μ0=1/c2.
Picture credit: Geek3, from Wikipedia. Licensed under CC-3.0 Attribution Share-Alike.

Many people have heard of Maxwell’s famous four equations:

∇·E = 0
∇·B = 0
∇ x E = -∂B/∂t
∇ x B = μ0J + μ0ε0(∂E/∂t)

But did you know that they can actually all be captured in one simple expression? When we extend classical electrodynamics into Minkowski 4-space (the geometry of special relativity), the time derivatives that appear in the derivation of Maxwell’s equations can be mixed into the same mathematics that takes care of the spatial expressions. By generalizing the Laplacian (∇2) to 4-space, we obtain the d’Alembertian (□2, sometimes written without the exponent). This operator allows us to rewrite Maxwell’s equations in tensor notation as one simple formula, sometimes called the Riemann–Sommerfeld equation:

2Aμ = -μ0Jμ

Where Aμ is the four-potential (φ, cA) (with φ and A the familiar scalar and vector potentials) and Jμ the four-current (ρc, J) (with ρ and J the familiar charge and current densities), thereby using the power of differential geometry and tensor notation to capture all of classical electrodynamics in one swift stroke.


Background

Maxwell’s Equations describe how electric and magnetic fields relate to each other and how they relate to the presence of charged particles and currents. The way we’ve presented them above is in differential form, which describes how electric fields (E) and magnetic fields (B) change in space and time; they can also be presented in integral form, which describes how the total measure of these fields scales with quantities like current density (J).

Although these equations were originally formulated in terms of fields, physicists would later find that the equations could be much cleaner when expressed in terms of potentials. A similar revolution happened in classical mechanics; although Newton’s forces were a good start, Lagrangian dynamics showed us that understanding the potential energy of the system made solving for equations of motion much simpler than trying to add up dozens of force vectors. In electrodynamics, we have two kinds of potentials: the scalar potential φ (or, sometimes, V), which most people know by the term “voltage”; and the vector potential, A, which is related to magnetic fields and may be less familiar.

Tensors notation (for our purposes) is little more than a convenient way to represent vectors and matrices. When you read the Riemann-Sommerfeld equation, then, you’re really reading 4 equations at once. What are those equations? Repeat the equation four times but each time replace μ with a different value; so, one equation might be □2A1 = -μ0J1. The 1’s in this equation aren’t exponents, but rather tensor indices; just like you might see vector components written as vx, vy, and vz representing different components of v, A0, A1, A2, and A3 represent the four components of the (vector) tensor A.

What are the funny upside-down triangles? The symbol is called a nabla, and is often read as “del”. In some ways, it’s an abuse of notation, but it gives us a convenient way to write the gradient, divergence, curl, and Laplacian operators. The exercise for how this notation works is left to the reader, but note the definitions: ∇U = grad(U), the gradient; ∇·U = div(U), the divergence; ∇ x U = curl(U), the curl; and ∇2U = ∇·(∇U) = div(grad(U)), the Laplacian. Each of these functions has a very important place in vector analysis. Since we represent electric and magnetic fields mathematically as vector fields, these operations give us information about the geometry of those fields. The gradient function tells us the direction that a field changes most at each point; the divergence tells us how much the field is dispersing (a field where everything pointed away from one center point would have a high divergence); the curl tells us how much a field is swirling, so to speak; and the Laplacian tells us how much a field changes in strength as it disperses.

And finally, we have our partial derivatives (∂B/∂t, for example) that give us rates of change (like how a magnetic field B changes with respect to the time t), and our constants, ε0 and μ0. These constants have the interesting relation that, for the speed of light c, ε0μ0=1/c2.

Picture credit: Geek3, from Wikipedia. Licensed under CC-3.0 Attribution Share-Alike.

Monday, July 11, 2011

Sizes of Infinity

Everyone knows about infinity, but most people don’t know that there are actually different sizes of infinity: the transfinite numbers. This may out odd at first; after all, how do you define the size of something that is infinite?

For example: there are an infinite number of even numbers. But, to be more precise, there is a “countable infinity” of even numbers; in other words, you can enumerate them.

Say we start at 0 and call it the first even number; then we say that 2 is the second even number, -2 is the third, 4 is the fourth, -4 is the fifth, et cetera. In this way we can assign an integer to each even number. (Note that we can choose any numbering scheme we like; what’s important isn’t how we enumerate them, but rather that we can.) Therefore, the size of the even numbers is as big as the size of the integers, and that size is “countable infinity”, often represented by the symbol aleph zero. A similar argument can be given for odd numbers, positive numbers, negative numbers, primes, and so forth.

On the other hand: when one tries to count the real numbers, a problem arises. For any two real numbers there will always exist another between them. It is not possible to create a one-to-one correspondence between the real numbers and the integers; in other words, they are impossible to count. For a rigorous proof of this fact see Cantor’s diagonal argument.

Notice that between any two rational numbers (numbers of the form a/b) there is also always another number; nevertheless, we can construct a an isomorphism between the rationals and the integers. So, in fact, there are just as many rationals as there are integers! One such isomorphism is the Calkin-Wilf tree.