Directional Derivative, Gradient

1. Suppose we have a scalar function of several variables
(1)\displaystyle  f\colon{\mathbb R}^{3}\rightarrow{\mathbb R}
Let {\widehat{u}} be some unit vector. How does {f} change in the {\widehat{u}} direction?

We can consider this quantity as a function
(2)\displaystyle  g(\vec{x}) = \lim_{h\rightarrow0}\frac{f(\vec{x}+h\widehat{u})-f(\vec{x})}{h}
What does this look like? Continue reading

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Chain Rule for Partial Derivatives

1. Problem. Consider a function {f(x,y)} where we parametrize
(1)\displaystyle  x=x(t,u),\quad\mbox{and}\quad y=y(t,u).
If {t\rightarrow t+\Delta t}, how does {f\rightarrow f+\Delta f} change? We will need to use partial derivatives and Taylor expansionContinue reading

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Curves, Velocity, “Classical Kinematics”

1. Curves. We are interested in describing the motion of my car. Well, everyone is interested in the motion of my car. How can we describe it mathematically?

First we approximate the car as a point. The point-like car moves in time, so the value of its components are functions of time. More precisely, the position of my car is
(1)\displaystyle  \vec{r}(t) = \langle f(t),g(t),h(t)\rangle = f(t)\widehat{\textbf{\i}} + g(t)\widehat{\textbf{\j}} + h(t)\widehat{\textbf{k}}
where the functions {f(t)}, {g(t)}, and {h(t)} are sometimes called component functions. Another way to think about this is writing
(2)\displaystyle  \vec{r}\colon[0,1]\rightarrow{\mathbb R}^{3}
where {0\leq t\leq1}.

Classical mechanics studies such curves under various circumstances. We will discuss some notions of kinematics, and study what it means to differentiate curves. Continue reading

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“High School Mathematics”: Naive Foundations

1. It dawned on me I should specify what exactly I mean by “high school mathematics” mentioned in the “About” page.

Briefly: it’s all the mathematics we learned in secondary school (well, American secondary school) plus some naive set theory. Set theoretic considerations aside, we will explicitly state the operators and axioms used.

I may revise this post, later on, to use Knuth’s “naive” construction of the real numbers using integers, just to be more explicit. Continue reading

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Continuity for Functions of Several Variables, Partial Derivatives

1. We considered differentiating and integrating functions of a single-variable. How? We began with the notion of a limit, and then considered the derivative. If we have a, e.g., polynomial
(1)\displaystyle  p(x,y) = x^{3}+x^{2}y+xy^{2}+y^{3}
we see
(2)\displaystyle  p(x+\Delta x,y) = x^{3}+x^{2}y+xy^{2}+y^{3}+\bigl(3x^{2}+2xy+y\bigr)\Delta x+ \mathcal{O}(\Delta x^{2})
Again we stop and reflect: this treats {y} as if it were constant. So the derivative formed by
(3)\displaystyle  \lim_{\Delta x\rightarrow0}\frac{p(x+\Delta x,y)-p(x,y)}{\Delta x}=3x^{2}+2xy+y
are “incomplete” or partial. There is some subtlety here due to using multiple variables, and we have to discuss the problems of limits first. Continue reading

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Surfaces (…well, “Quadrics”)

1. Let {w=f(x_{1},x_{2},\dots,x_{n})} where {x_{1}}, {x_{2}}, …, {x_{n}} are all independent variables. Then the “Domain” of {f} is the set of {n}-tuples {{\mathbb R}^{n}}. Note that an ordered pair is
(1)\displaystyle  (x,y)=\mbox{2-tuple}.
The set of corresponding values is the “Range” (or Codomain) of the function. So we have
(2)\displaystyle  f\colon\underbrace{{\mathbb R}^{n}}_{\text{domain}}\rightarrow\underbrace{{\mathbb R}}_{\text{range}}
Sometimes we write {\mathrm{dom}(f)} for the domain of {f}, and {\mathrm{ran}(f)} or {\mathrm{cod}(f)} for the range (or codomain) of {f}. We DO NOT write {f({\mathbb R}^{2})} for the range, because this is the collection of all points mapped by {f}.

Example 1. Consider {f\colon{\mathbb R}^{2}\rightarrow{\mathbb R}} defined by
(3)\displaystyle  f(x,y) = \sqrt{x^{2}+y^{2}}
Notice that {f(x,y)\geq0} for any {x,y\in{\mathbb R}}. So
(4)\displaystyle  f({\mathbb R}^{2})=\{f(x,y)\in{\mathbb R} : x,y\in{\mathbb R}\} = \{ u\in{\mathbb R} : u\geq0\}\not={\mathbb R}.
This is not the codomain! It’s contained in the codomain, though. The image is
always a subset of the codomain.

Continue reading

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Constructing Planes

1. We can use vectors and the cross product to construct planes quite easily. Continue reading

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