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A.4: Table of Integrals
https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/04%3A_Appendices/4.01%3A_A_Appendices/4.1.04%3A_A.4%3A_Table_of_Integrals
Throughout this table, $a$ and $b$ are given constants, independent of $x$ and $C$ is an arbitrary constant. $a\int f(x)\ \mathrm{d}{x} +b\int g(x)\ \mathrm{d}{x} \ +\ C$ \(\int f(x)\ \mathr...Throughout this table, $a$ and $b$ are given constants, independent of $x$ and $C$ is an arbitrary constant. $a\int f(x)\ \mathrm{d}{x} +b\int g(x)\ \mathrm{d}{x} \ +\ C$ $\int f(x)\ \mathrm{d}{x} +\int g(x)\ \mathrm{d}{x} \ +\ C$ $\int f(x)\ \mathrm{d}{x} -\int g(x)\ \mathrm{d}{x} \ +\ C$ $a\int f(x)\ \mathrm{d}{x} \ +\ C$ $\ln |\csc x-\cot x|+C$ $\ln |\sec x+\tan x|+C$ $\frac{1}{\ln a}\ a^x+C$ $x\ln x -x+C$ $\textrm{arcsec} x+C$ \quad( $x \gt 1$ )
2.4: The Chain Rule
https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/02%3A_Partial_Derivatives/2.04%3A_The_Chain_Rule
You already routinely use the one dimensional chain rule
A.3: Table of Derivatives
https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/04%3A_Appendices/4.01%3A_A_Appendices/4.1.03%3A_A.3%3A_Table_of_Derivatives
$af'(x)+bg'(x)$ $f'(x)+g'(x)$ $f'(x)-g'(x)$ $f'(x)g(x)+f(x)g'(x)$ $f'(x)g(x)h(x)+f(x)g'(x)h(x)+f(x)g(x)h'(x)$ $\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}$ $-\frac{g'(x)}{g(x)^2}$ \(f'\big(g(x)\b... $af'(x)+bg'(x)$ $f'(x)+g'(x)$ $f'(x)-g'(x)$ $f'(x)g(x)+f(x)g'(x)$ $f'(x)g(x)h(x)+f(x)g'(x)h(x)+f(x)g(x)h'(x)$ $\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}$ $-\frac{g'(x)}{g(x)^2}$ $f'\big(g(x)\big)g'(x)$ $x^a$ $g'(x)\cos g(x)$ $-g'(x)\sin g(x)$ $a^x$ $(\ln a)\ a^x$ $F'(x)=\frac{\mathrm{d}F}{\mathrm{d}x}$ $\frac{1}{x}$ $\frac{g'(x)}{g(x)}$ $\frac{1}{1+x^2}$ $\frac{g'(x)}{1+g(x)^2}$ $-\frac{1}{|x|\sqrt{x^2-1}}$ $\frac{1}{|x|\sqrt{x^2-1}}$ $-\frac{1}{1+x^2}$
A.7: ISO Coordinate System Notation
https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/04%3A_Appendices/4.01%3A_A_Appendices/4.1.07%3A_A.7%3A_ISO_Coordinate_System_Notation
In this text we have chosen symbols for the various polar, cylindrical and spherical coordinates that are standard for mathematics. There is another, different, set of symbols that are commonly used i...In this text we have chosen symbols for the various polar, cylindrical and spherical coordinates that are standard for mathematics. There is another, different, set of symbols that are commonly used in the physical sciences and engineering. Indeed, there is an international convention, called ISO 80000-2, that specifies those symbols. In this appendix, we summarize the definitions and standard properties of the polar, cylindrical and spherical coordinate systems using the ISO symbols.
2.7: Directional Derivatives and the Gradient
https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/02%3A_Partial_Derivatives/2.07%3A_Directional_Derivatives_and_the_Gradient
The principal interpretation of $\frac{\mathrm{d}f}{\mathrm{d}x}(a)$ is the rate of change of $f(x)\text{,}$ per unit change of $x\text{,}$ at $x=a\text{.}$ The natural analog of this interpre...The principal interpretation of $\frac{\mathrm{d}f}{\mathrm{d}x}(a)$ is the rate of change of $f(x)\text{,}$ per unit change of $x\text{,}$ at $x=a\text{.}$ The natural analog of this interpretation for multivariable functions is the directional derivative, which we now introduce through a question.
2.8: Optional — Solving the Wave Equation
https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/02%3A_Partial_Derivatives/2.08%3A_Optional__Solving_the_Wave_Equation
Many phenomena are modelled by equations that relate the rates of change of various quantities. As rates of change are given by derivatives the resulting equations contain derivatives and so are calle...Many phenomena are modelled by equations that relate the rates of change of various quantities. As rates of change are given by derivatives the resulting equations contain derivatives and so are called differential equations.
1.4: Equations of Planes in 3d
https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/01%3A_Vectors_and_Geometry_in_Two_and_Three_Dimensions/1.04%3A_Equations_of_Planes_in_3d
Specifying one point $(x_0,y_0,z_0)$ on a plane and a vector $\textbf{d}$ parallel to the plane does not uniquely determine the plane, because it is free to rotate about $\textbf{d}\text{.}$ \vd\...Specifying one point $(x_0,y_0,z_0)$ on a plane and a vector $\textbf{d}$ parallel to the plane does not uniquely determine the plane, because it is free to rotate about $\textbf{d}\text{.}$ \vd\text{.}
3.1: Double Integrals
https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/03%3A_Multiple_Integrals/3.01%3A_Double_Integrals
Suppose that you want to compute the mass of a plate that fills the region $\mathcal{R}$ in the $xy$ -plane. Suppose further that the density of the plate, say in kilograms per square meter, depend...Suppose that you want to compute the mass of a plate that fills the region $\mathcal{R}$ in the $xy$ -plane. Suppose further that the density of the plate, say in kilograms per square meter, depends on position.
1: Vectors and Geometry in Two and Three Dimensions
https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/01%3A_Vectors_and_Geometry_in_Two_and_Three_Dimensions
Before we get started doing calculus in two and three dimensions we need to brush up on some basic geometry, that we will use a lot. We are already familiar with the Cartesian plane 1 , but we'll star...Before we get started doing calculus in two and three dimensions we need to brush up on some basic geometry, that we will use a lot. We are already familiar with the Cartesian plane 1 , but we'll start from the beginning. René Descartes (1596–1650) was a French scientist and philosopher, who lived in the Dutch Republic for roughly twenty years after serving in the (mercenary) Dutch States Army. He is viewed as the father of analytic geometry, which uses numbers to study geometry.
2.5: Tangent Planes and Normal Lines
https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/02%3A_Partial_Derivatives/2.05%3A_Tangent_Planes_and_Normal_Lines
The tangent line to the curve $y=f(x)$ at the point $\big(x_0,f(x_0)\big)$ is the straight line that fits the curve best at that point.
A.6: 3-D Coordinate Systems
https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/04%3A_Appendices/4.01%3A_A_Appendices/4.1.06%3A_A.6%3A_3d_Coordinate_Systems
\[\begin{align*} \rho&=\text{ distance from }(0,0,0)\text{ to }(x,y,z)\\ \vec{a}rphi&=\text{ angle between the $z$ axis and the line joining $(x,y,z)$ to $(0,0,0)$}\\ \theta&=\text{ angle between the ... $\begin{align*} \rho&=\text{ distance from }(0,0,0)\text{ to }(x,y,z)\\ \vec{a}rphi&=\text{ angle between the $z$ axis and the line joining $(x,y,z)$ to $(0,0,0)$}\\ \theta&=\text{ angle between the $x$ axis and the line joining $(x,y,0)$ to $(0,0,0)$} \end{align*}$

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