{ "cross-product": { "sourcePage": "/appendix/cross-product/", "id": "cross-product", "html": "

The cross product of $\\mathbf{u} = (u_1, u_2, u_3)^\\top$ and $\\mathbf{v} = (v_1, v_2, v_3)^\\top$ in $\\mathbb{R}^3$ is:

$\\mathbf{u} \\times \\mathbf{v} = \\begin{pmatrix} u_2 v_3 - u_3 v_2 \\\\ u_3 v_1 - u_1 v_3 \\\\ u_1 v_2 - u_2 v_1 \\end{pmatrix}$

" }, "differentiation-rules": { "sourcePage": "/appendix/differentiation-rules/", "id": "differentiation-rules", "html": "

Let $f$ and $g$ be differentiable functions and $c \\in \\mathbb{R}$ a constant. The fundamental differentiation rules are:

$\\frac{d}{dx}[c] = 0 \\qquad \\frac{d}{dx}[x^n] = n x^{n-1}$

$\\frac{d}{dx}[c \\cdot f] = c \\cdot f' \\qquad \\frac{d}{dx}[f + g] = f' + g'$

$\\frac{d}{dx}[f \\cdot g] = f' g + f g' \\qquad \\frac{d}{dx}\\!\\left[\\frac{f}{g}\\right] = \\frac{f' g - f g'}{g^2}$

$\\frac{d}{dx}[f(g(x))] = f'(g(x)) \\cdot g'(x)$

" }, "dot-product": { "sourcePage": "/appendix/dot-product/", "id": "dot-product", "html": "

The dot product of $\\mathbf{u} = (u_1, \\ldots, u_n)^\\top$ and $\\mathbf{v} = (v_1, \\ldots, v_n)^\\top$ in $\\mathbb{R}^n$ is:

$\\langle \\mathbf{u}, \\mathbf{v} \\rangle = \\mathbf{u} \\cdot \\mathbf{v} = \\mathbf{u}^\\top \\mathbf{v} = \\sum_{i=1}^{n} u_i v_i$

" }, "dot-product-geometric": { "sourcePage": "/appendix/dot-product/", "id": "dot-product-geometric", "html": "

For $\\mathbf{u}, \\mathbf{v} \\in \\mathbb{R}^n$ , the dot product equals:

$\\langle \\mathbf{u}, \\mathbf{v} \\rangle = \\|\\mathbf{u}\\| \\, \\|\\mathbf{v}\\| \\cos \\theta$

where $\\theta \\in [0, \\pi]$ is the angle between the two vectors and $\\|\\cdot\\|$ denotes the Euclidean norm.

" }, "standard-basis": { "sourcePage": "/appendix/standard-basis/", "id": "standard-basis", "html": "

The standard basis of $\\mathbb{R}^n$ is the set $\\{\\mathbf{e}_1, \\ldots, \\mathbf{e}_n\\}$ , where each unit coordinate vector (also known as a standard unit vector or coordinate unit vector) $\\mathbf{e}_i$ has a $1$ in position $i$ and $0$ everywhere else:

$\\mathbf{e}_i = (0, \\ldots, 0, \\underbrace{1}_{i\\text{-th slot}}, 0, \\ldots, 0)^\\top, \\quad i \\in \\{1, \\ldots, n\\}$

" }, "trigonometric-derivatives": { "sourcePage": "/appendix/trigonometric-derivatives/", "id": "trigonometric-derivatives", "html": "

The derivatives of the trigonometric functions are:

$\\frac{d}{dx} \\sin x = \\cos x \\qquad \\frac{d}{dx} \\cos x = -\\sin x \\qquad \\frac{d}{dx} \\tan x = \\frac{1}{\\cos^2 x} = \\sec^2 x$

$\\frac{d}{dx} \\cot x = -\\frac{1}{\\sin^2 x} = -\\csc^2 x \\qquad \\frac{d}{dx} \\sec x = \\sec x \\tan x \\qquad \\frac{d}{dx} \\csc x = -\\csc x \\cot x$

" }, "multivariate-function": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "multivariate-function", "html": "

A multivariate function takes a vector of $n$ inputs and produces a vector of $m$ outputs:

$f : D \\subseteq \\mathbb{R}^n \\to \\mathbb{R}^m, \\quad \\mathbf{x} = \\begin{pmatrix} x_1 \\\\ \\vdots \\\\ x_n \\end{pmatrix} \\mapsto f(\\mathbf{x}) = \\begin{pmatrix} f_1(x_1, \\ldots, x_n) \\\\ \\vdots \\\\ f_m(x_1, \\ldots, x_n) \\end{pmatrix}$

" }, "plane-curve": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "plane-curve", "html": "

A plane curve takes a single real parameter $t \\in D$ and produces a point in 2D space, tracing a path through the plane as $t$ varies:

$\\gamma : D \\subseteq \\mathbb{R} \\to \\mathbb{R}^2 \\quad (n = 1,\\ m = 2)$

" }, "space-curve": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "space-curve", "html": "

A space curve takes a single real parameter $t \\in D$ and produces a point in 3D space, tracing a path through three-dimensional space as $t$ varies:

$\\gamma : D \\subseteq \\mathbb{R} \\to \\mathbb{R}^3 \\quad (n = 1,\\ m = 3)$

" }, "parametric-surface": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "parametric-surface", "html": "

A parametric surface is defined by a map $(u, v) \\in D \\mapsto \\mathbb{R}^3$ that takes two free parameters and produces a point in 3D space. As $(u, v)$ varies continuously over the domain $D$ , the output traces out a 2D surface embedded in 3D. Unlike an explicit surface $z = f(x, y)$ — where the surface is directly the graph of a function — a parametric surface uses two auxiliary parameters with no inherent geometric meaning, which makes it possible to describe shapes like spheres or tori that cannot be expressed as a single function of $x$ and $y$ .

$\\phi : D \\subseteq \\mathbb{R}^2 \\to \\mathbb{R}^3 \\quad (n = 2,\\ m = 3)$

The two input parameters act like coordinates on the surface itself — by varying them over the domain $D$ , we trace out every point on the surface.

" }, "scalar-field": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "scalar-field", "html": "

A scalar field assigns a single real number to each point in an $n$ -dimensional space:

$f : D \\subseteq \\mathbb{R}^n \\to \\mathbb{R} \\quad (n \\in \\mathbb{N},\\ m = 1)$

" }, "vector-field": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "vector-field", "html": "

A vector field assigns a vector to each point in space:

$\\mathbf{v} : D \\subseteq \\mathbb{R}^n \\to \\mathbb{R}^n \\quad (n = m,\\ n \\in \\mathbb{N})$

" }, "euclidean-norm": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "euclidean-norm", "html": "

The Euclidean norm $\\|\\cdot\\|$ is the standard \"ruler distance\" — it measures the straight-line length (magnitude) from the origin to a coordinate point $\\mathbf{x} = (x_1, \\ldots, x_n)^\\top \\in \\mathbb{R}^n$ :

$\\|(x_1, \\ldots, x_n)^\\top\\| = \\sqrt{x_1^2 + \\cdots + x_n^2}$

" }, "epsilon-ball": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "epsilon-ball", "html": "

The $\\varepsilon$ -ball of radius $\\varepsilon > 0$ centered at $\\mathbf{x}_0$ is the set of all points within distance $\\varepsilon$ of $\\mathbf{x}_0$ :

$B_\\varepsilon(\\mathbf{x}_0) = \\{\\, \\mathbf{x} \\in \\mathbb{R}^n \\mid \\|\\mathbf{x} - \\mathbf{x}_0\\| < \\varepsilon \\,\\}$

" }, "inner-point": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "inner-point", "html": "

A point $\\mathbf{x}_0 \\in D$ is called an inner point of $D$ if there exists some $\\varepsilon > 0$ such that $B_\\varepsilon(\\mathbf{x}_0) \\subseteq D$ — that is, we can place an open ball around $\\mathbf{x}_0$ that fits entirely within $D$ , without touching or crossing its boundary.

" }, "interior": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "interior", "html": "

The set of all inner points of $D$ is called the interior of $D$ , denoted $\\mathring{D}$ .

" }, "open-set": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "open-set", "html": "

A set $D \\subseteq \\mathbb{R}^n$ is called open if it equals its own interior: $D = \\mathring{D}$

" }, "boundary-point": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "boundary-point", "html": "

A point $\\mathbf{x}_0 \\in \\mathbb{R}^n$ is called a boundary point of $D$ if every open ball around it, no matter how small, contains points both inside and outside $D$ :

$B_\\varepsilon(\\mathbf{x}_0) \\cap D \\neq \\emptyset \\quad \\text{and} \\quad B_\\varepsilon(\\mathbf{x}_0) \\cap D^c \\neq \\emptyset \\qquad \\text{for all } \\varepsilon > 0.$

" }, "boundary": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "boundary", "html": "

The set of all boundary points of $D$ is called the boundary of $D$ , denoted $\\partial D$ .

" }, "closure": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "closure", "html": "

The closure of $D$ is the set $\\bar{D} = D \\cup \\partial D$ — that is, $D$ together with all of its boundary points.

" }, "closed-set": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "closed-set", "html": "

A set $D \\subseteq \\mathbb{R}^n$ is called closed if it contains all of its boundary points: $\\partial D \\subseteq D$ , or equivalently, $\\bar{D} = D$ .

" }, "bounded-set": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "bounded-set", "html": "

A set $D \\subseteq \\mathbb{R}^n$ is called bounded if there exists some $K \\in \\mathbb{R}$ such that $\\|\\mathbf{x}\\| < K$ for all $\\mathbf{x} \\in D$ .

" }, "compact-set": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "compact-set", "html": "

A set $D \\subseteq \\mathbb{R}^n$ is called compact if it is both closed and bounded.

" }, "convex-set": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "convex-set", "html": "

A set $D \\subseteq \\mathbb{R}^n$ is called convex if for any two points $\\mathbf{x}, \\mathbf{y} \\in D$ , the entire line segment connecting them also lies in $D$ :

$\\forall\\, \\mathbf{x}, \\mathbf{y} \\in D :\\ \\alpha\\mathbf{x} + (1 - \\alpha)\\mathbf{y} \\in D \\qquad \\text{for all } 0 \\leq \\alpha \\leq 1.$

" }, "vector-sequence-convergence": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "vector-sequence-convergence", "html": "

Convergence of a Vector Sequence

A sequence of vectors $\\left(\\mathbf{x}^{(k)}\\right)_{k \\in \\mathbb{N}_0}$ converges to the limit vector $\\mathbf{x}$ if the distance between $\\mathbf{x}^{(k)}$ and $\\mathbf{x}$ vanishes as $k \\to \\infty$ :

$\\lim_{k \\to \\infty} \\| \\mathbf{x}^{(k)} - \\mathbf{x} \\| = 0$

" }, "continuity-vector-function": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/functions-of-several-variables/", "id": "continuity-vector-function", "html": "

Given a vector function $f : D \\subseteq \\mathbb{R}^n \\to \\mathbb{R}^m$ , we say that $f$ is:

\n
continuous at $\\mathbf{a} \\in D$ if, for every sequence $\\left(\\mathbf{x}^{(k)}\\right)_{k \\in \\mathbb{N}_0}$ in $D$ with $\\mathbf{x}^{(k)} \\to \\mathbf{a}$ , the corresponding output sequence $\\left(f\\left(\\mathbf{x}^{(k)}\\right)\\right)_{k \\in \\mathbb{N}_0}$ in $\\mathbb{R}^m$ converges to $f(\\mathbf{a})$ . In short, there is no \"teleporting\" allowed. As the input smoothly approaches a specific destination $\\mathbf{a}$ , the output must smoothly approach the actual output at that destination $f(\\mathbf{a})$ without sudden jumps or glitches.
\n
\n
continuous on $D$ if it is continuous at every point $\\mathbf{a} \\in D$ .
\n

" }, "directional-derivative": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/partial-differentiation/", "id": "directional-derivative", "html": "

The directional derivative of a scalar field $f : D \\subseteq \\mathbb{R}^n \\to \\mathbb{R}$ at a point $\\mathbf{a} \\in D$ in the direction of a unit vector $\\mathbf{v}$ (with $\\|\\mathbf{v}\\| = 1$ ) is:

$\\frac{\\partial f}{\\partial \\mathbf{v}}(\\mathbf{a}) = \\partial_\\mathbf{v} f(\\mathbf{a}) = f_\\mathbf{v}(\\mathbf{a}) = \\lim_{h \\to 0} \\frac{f(\\mathbf{a} + h\\mathbf{v}) - f(\\mathbf{a})}{h}$

" }, "partial-derivative": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/partial-differentiation/", "id": "partial-derivative", "html": "

The partial derivative of $f : D \\subseteq \\mathbb{R}^n \\to \\mathbb{R}$ at a point $\\mathbf{a} \\in D$ with respect to the variable $x_i$ is:

$\\frac{\\partial f}{\\partial x_i}(\\mathbf{a}) = \\partial_i f(\\mathbf{a}) = f_{x_i}(\\mathbf{a}) = \\lim_{h \\to 0} \\frac{f(\\mathbf{a} + h\\mathbf{e}_i) - f(\\mathbf{a})}{h}, \\quad i \\in \\{1, \\ldots, n\\}$

" }, "gradient": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/partial-differentiation/", "id": "gradient", "html": "

The gradient of $f : D \\subseteq \\mathbb{R}^n \\to \\mathbb{R}$ at a point $\\mathbf{a} \\in D$ is the column vector of all partial derivatives:

$\\nabla f(\\mathbf{a}) = \\begin{pmatrix} f_{x_1}(\\mathbf{a}) \\\\ \\vdots \\\\ f_{x_n}(\\mathbf{a}) \\end{pmatrix} = \\operatorname{grad} f(\\mathbf{a})$

" }, "isoline": { "sourcePage": "/chapters/focus-analysis-calculus/foundations/partial-differentiation/", "id": "isoline", "html": "

Let $f : D \\subseteq \\mathbb{R}^n \\to \\mathbb{R}$ be a scalar field. For a value $c \\in f(D) \\subseteq \\mathbb{R}$ , the isoline (also called contour line or level set) of $f$ at level $c$ is:

$N_c = \\{\\, \\mathbf{x} \\in D \\mid f(\\mathbf{x}) = c \\,\\}$

" } }

Roots and Optima