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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_and_Graph_Theory_(Guichard)/01%3A_Fundamentals/1.02%3A_ExamplesSuppose we have a chess board, and a collection of tiles, like dominoes, each of which is the size of two squares on the chess board. First we need to be clear on the rules: the board is covered if th...Suppose we have a chess board, and a collection of tiles, like dominoes, each of which is the size of two squares on the chess board. First we need to be clear on the rules: the board is covered if the dominoes are laid down so that each covers exactly two squares of the board; no dominoes overlap; and every square is covered. To make the problem more interesting, we allow the board to be rectangular of any size, and we allow some squares to be removed from the board.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/13%3A_Vector_Functions/13.03%3A_Arc_length_and_CurvatureSometimes it is useful to compute the length of a curve in space; for example, if the curve represents the path of a moving object, the length of the curve between two points may be the distance trave...Sometimes it is useful to compute the length of a curve in space; for example, if the curve represents the path of a moving object, the length of the curve between two points may be the distance traveled by the object between two times.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/15%3A_Multiple_Integration/15.01%3A_Volume_and_Average_HeightConsider a surface f(x,y); you might temporarily think of this as representing physical topography---a hilly landscape, perhaps. What is the average height of the surface (or average altitude of the ...Consider a surface f(x,y); you might temporarily think of this as representing physical topography---a hilly landscape, perhaps. What is the average height of the surface (or average altitude of the landscape) over some region? Multiple integration approach can be used to estimate these values.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/03%3A_Rules_for_Finding_Derivatives/3.01%3A_The_Power_RuleThe power rule addresses the derivative of a power function.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/15%3A_Multiple_Integration/15.02%3A_Double_Integrals_in_Cylindrical_CoordinatesHow might we approximate the volume under a surface in a way that uses cylindrical coordinates directly? The basic idea is the same as before: we divide the region into many small regions, multiply th...How might we approximate the volume under a surface in a way that uses cylindrical coordinates directly? The basic idea is the same as before: we divide the region into many small regions, multiply the area of each small region by the height of the surface somewhere in that little region, and add them up. What changes is the shape of the small regions
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/07%3A_Integration/7.01%3A_Two_ExamplesUp to now we have been concerned with extracting information about how a function changes from the function itself. Given knowledge about an object's position, for example, we want to know the object'...Up to now we have been concerned with extracting information about how a function changes from the function itself. Given knowledge about an object's position, for example, we want to know the object's speed. Given information about the height of a curve we want to know its slope. We now consider problems that are, whether obviously or not, the reverse of such problems.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/16%3A_Vector_Calculus/16.02%3A_Line_IntegralsWe have so far integrated "over'' intervals, areas, and volumes with single, double, and triple integrals. We now investigate integration over or "along'' a curve---"line integrals'' are really "curve...We have so far integrated "over'' intervals, areas, and volumes with single, double, and triple integrals. We now investigate integration over or "along'' a curve---"line integrals'' are really "curve integrals''.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/12%3A_Three_Dimensions/12.01%3A_The_Coordinate_SystemSo far we have been investigating functions with one independent and one dependent variable. Such functions can be represented in two dimensions, using two numerical axes that allow us to identify eve...So far we have been investigating functions with one independent and one dependent variable. Such functions can be represented in two dimensions, using two numerical axes that allow us to identify every point in the plane with two numbers. We now want to talk about three-dimensional space; to identify every point in three dimensions we require three numerical values. The obvious way to make this association is to add one new axis, perpendicular to the x and y axes we already understand.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/17%3A_Differential_EquationsDifferential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and en...Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/14%3A_Partial_Differentiation/14.05%3A_Directional_DerivativesThe directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x...The directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/06%3A_Applications_of_the_Derivative/6.02%3A_Related_RatesIn differential calculus, related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is...In differential calculus, related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time.
