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60-615: The length of the Bazaikha is 128 kilometres (80 mi) and the area of its basin is 21,700 square kilometres (8,400 sq mi). Beginning in the Eastern Sayan mountains it flows west-northward without a contact with populated places until the very confluence with the Yenisey in the suburbs near the southwestern edge of the Krasnoyarsk city. The valley is mostly deep and winding with

120-1770: A ϵ > 0 {\displaystyle \epsilon >0} such that | s ( t ) − s ( t 0 ) | < δ {\displaystyle |s(t)-s(t_{0})|<\delta } ∀ | t − t 0 | < ϵ {\displaystyle \forall |t-t_{0}|<\epsilon } . Hence, for every α > 0 {\displaystyle \alpha >0} , choose δ = α K {\displaystyle \delta ={\frac {\alpha }{K}}} ; there exists an ϵ > 0 {\displaystyle \epsilon >0} such that for all t {\displaystyle t} satisfying | t − t 0 | < ϵ {\displaystyle |t-t_{0}|<\epsilon } , | s ( t ) − s ( t 0 ) | < δ {\displaystyle |s(t)-s(t_{0})|<\delta } , and | f ( s ( t ) ) − f ( s ( t 0 ) ) | ≤ K | s ( t ) − s ( t 0 ) | < K δ = α {\displaystyle |f(s(t))-f(s(t_{0}))|\leq K|s(t)-s(t_{0})|<K\delta =\alpha } . Hence lim t → t 0 f ( s ( t ) ) {\displaystyle \lim _{t\to t_{0}}f(s(t))} converges to f ( s ( t 0 ) ) {\displaystyle f(s(t_{0}))} regardless of

180-430: A ≤ 1 {\displaystyle 0\leq a\leq 1} by are continuous. Specifically, However, consider the parametric path x ( t ) = t , y ( t ) = t {\displaystyle x(t)=t,\,y(t)=t} . The parametric function becomes Therefore, It is hence clear that the function is not multivariate continuous, despite being continuous in both coordinates. From

240-400: A definite integral : The formula for the area enclosed by an ellipse is related to the formula of a circle; for an ellipse with semi-major and semi-minor axes x and y the formula is: Most basic formulas for surface area can be obtained by cutting surfaces and flattening them out (see: developable surfaces ). For example, if the side surface of a cylinder (or any prism )

300-399: A linear transformation which directly varies from point to point in the domain of the function. Differential equations containing partial derivatives are called partial differential equations or PDEs. These equations are generally more difficult to solve than ordinary differential equations , which contain derivatives with respect to only one variable. The multiple integral extends

360-417: A corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres (m ), square centimetres (cm ), square millimetres (mm ), square kilometres (km ), square feet (ft ), square yards (yd ), square miles (mi ), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units. The SI unit of area

420-496: A direction; it is clear for example that ∇ u ^ f ( x 0 ) = − ∇ − u ^ f ( x 0 ) {\displaystyle \nabla _{\hat {\mathbf {u}}}f(x_{0})=-\nabla _{-{\hat {\mathbf {u}}}}f(x_{0})} . It is also possible for directional derivatives to exist for some directions but not for others. The partial derivative generalizes

480-422: A general limit at the point ( 0 , 0 ) {\displaystyle (0,0)} cannot be defined for the function. A general limit can be defined if the limits to a point along all possible paths converge to the same value, i.e. we say for a function f : R n → R m {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}} that

540-478: A path may be defined by considering a parametrised path s ( t ) : R → R n {\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}} in n-dimensional Euclidean space. Any function f ( x → ) : R n → R m {\displaystyle f({\overrightarrow {x}}):\mathbb {R} ^{n}\to \mathbb {R} ^{m}} can then be projected on

600-428: A rectangle with length l and width w , the formula for the area is: That is, the area of the rectangle is the length multiplied by the width. As a special case, as l = w in the case of a square, the area of a square with side length s is given by the formula: The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom . On

660-435: A single variable. In multivariate calculus, it is required to generalize these to multiple variables, and the domain is therefore multi-dimensional. Care is therefore required in these generalizations, because of two key differences between 1D and higher dimensional spaces: The consequence of the first difference is the difference in the definition of the limit and differentiation. Directional limits and derivatives define

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720-433: A solid shape such as a sphere , cone, or cylinder, the area of its boundary surface is called the surface area . Formulas for the surface areas of simple shapes were computed by the ancient Greeks , but computing the surface area of a more complicated shape usually requires multivariable calculus . Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area

780-521: A sphere was first obtained by Archimedes in his work On the Sphere and Cylinder . The formula is: where r is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to calculus . Multivariable calculus Multivariable calculus (also known as multivariate calculus ) is the extension of calculus in one variable to calculus with functions of several variables :

840-408: Is or, when expressed in terms of ordinary differentiation, which is a well defined expression because f ( x 0 + u ^ t ) {\displaystyle f(x_{0}+{\hat {\mathbf {u}}}t)} is a scalar function with one variable in t {\displaystyle t} . It is not possible to define a unique scalar derivative without

900-625: Is Lipschitz continuous at s ( t 0 ) {\displaystyle s(t_{0})} , and that the limit exits for at least one such path. For s ( t ) {\displaystyle s(t)} continuous up to the first derivative (this statement is well defined as s {\displaystyle s} is a function of one variable), we can write the Taylor expansion of s {\displaystyle s} around t 0 {\displaystyle t_{0}} using Taylor's theorem to construct

960-447: Is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram is r , and the width is half the circumference of the circle, or π r . Thus, the total area of the circle is π r : Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of

1020-418: Is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a cone , the side surface can be flattened out into a sector of a circle, and the resulting area computed. The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature , it cannot be flattened out. The formula for the surface area of

1080-398: Is easy to verify that this function is zero by definition on the boundary and outside of the quadrangle ( 0 , 1 ) × ( 0 , 1 ) {\displaystyle (0,1)\times (0,1)} . Furthermore, the functions defined for constant x {\displaystyle x} and y {\displaystyle y} and 0 ≤

1140-462: Is known as Heron's formula for the area of a triangle in terms of its sides, and a proof can be found in his book, Metrica , written around 60 CE. It has been suggested that Archimedes knew the formula over two centuries earlier, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work. In 300 BCE Greek mathematician Euclid proved that

1200-431: Is related to the definition of determinants in linear algebra , and is a basic property of surfaces in differential geometry . In analysis , the area of a subset of the plane is defined using Lebesgue measure , though not every subset is measurable if one supposes the axiom of choice. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through

1260-470: Is the square metre, which is considered an SI derived unit . Calculation of the area of a square whose length and width are 1 metre would be: 1 metre × 1 metre = 1 m and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as: 3 metres × 2 metres = 6 m . This is equivalent to 6 million square millimetres. Other useful conversions are: In non-metric units,

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1320-455: The Cartesian coordinates ( x i , y i ) {\displaystyle (x_{i},y_{i})} ( i =0, 1, ..., n -1) of whose n vertices are known, the area is given by the surveyor's formula : where when i = n -1, then i +1 is expressed as modulus n and so refers to 0. The most basic area formula is the formula for the area of a rectangle . Given

1380-421: The differentiation and integration of functions involving multiple variables ( multivariate ), rather than just one. Multivariable calculus may be thought of as an elementary part of calculus on Euclidean space . The special case of calculus in three dimensional space is often called vector calculus . In single-variable calculus, operations like differentiation and integration are made to functions of

1440-478: The hectare is still commonly used to measure land: Other uncommon metric units of area include the tetrad , the hectad , and the myriad . The acre is also commonly used to measure land areas, where An acre is approximately 40% of a hectare. On the atomic scale, area is measured in units of barns , such that: The barn is commonly used in describing the cross-sectional area of interaction in nuclear physics . In South Asia (mainly Indians), although

1500-480: The length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). Two different regions may have the same area (as in squaring the circle ); by synecdoche , "area" sometimes is used to refer to the region, as in a " polygonal area ". The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI),

1560-419: The line integral are used to integrate over curved manifolds such as surfaces and curves . In single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral. The link between the derivative and the integral in multivariable calculus is embodied by the integral theorems of vector calculus: In a more advanced study of multivariable calculus, it

1620-417: The surveyor's formula for the area of any polygon with known vertex locations by Gauss in the 19th century. The development of integral calculus in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an ellipse and the surface areas of various curved three-dimensional objects. For a non-self-intersecting ( simple ) polygon,

1680-556: The 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates , but did not identify the constant of proportionality . Eudoxus of Cnidus , also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared. Subsequently, Book I of Euclid's Elements dealt with equality of areas between two-dimensional figures. The mathematician Archimedes used

1740-490: The Lipschitz continuity condition for f {\displaystyle f} we have where K {\displaystyle K} is the Lipschitz constant. Note also that, as s ( t ) {\displaystyle s(t)} is continuous at t 0 {\displaystyle t_{0}} , for every δ > 0 {\displaystyle \delta >0} there exists

1800-466: The area of a cyclic quadrilateral (a quadrilateral inscribed in a circle) in terms of its sides. In 1842, the German mathematicians Carl Anton Bretschneider and Karl Georg Christian von Staudt independently found a formula, known as Bretschneider's formula , for the area of any quadrilateral. The development of Cartesian coordinates by René Descartes in the 17th century allowed the development of

1860-399: The area of a shape or planar lamina , while surface area refers to the area of an open surface or the boundary of a three-dimensional object . Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of

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1920-501: The area of a triangle is half that of a parallelogram with the same base and height in his book Elements of Geometry . In 499 Aryabhata , a great mathematician - astronomer from the classical age of Indian mathematics and Indian astronomy , expressed the area of a triangle as one-half the base times the height in the Aryabhatiya . In the 7th century CE, Brahmagupta developed a formula, now known as Brahmagupta's formula , for

1980-405: The areas of simple shapes such as triangles , rectangles , and circles . Using these formulas, the area of any polygon can be found by dividing the polygon into triangles . For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus . For

2040-417: The areas of the approximate parallelograms is exactly π r , which is the area of the circle. This argument is actually a simple application of the ideas of calculus . In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to integral calculus . Using modern methods, the area of a circle can be computed using

2100-458: The banks covered with conifers. Numerous resorts, dachas and cottages are built in the valley stretching for 14 kilometres (9 mi) inwards from the confluence. By the Yenisei, the village of Bazaikha is located in what is considered a part of Krasnoyarsk; it should not be confused with the railway station named Bazaikha which is 18 kilometres (11 mi) west of it. The closest approach to

2160-1004: The concept of limit along a path, we can then derive the definition for multivariate continuity in the same manner, that is: we say for a function f : R n → R m {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}} that f {\displaystyle f} is continuous at the point x 0 {\displaystyle x_{0}} , if and only if for all continuous functions s ( t ) : R → R n {\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}} such that s ( t 0 ) = x 0 {\displaystyle s(t_{0})=x_{0}} . As with limits, being continuous along one path s ( t ) {\displaystyle s(t)} does not imply multivariate continuity. Continuity in each argument not being sufficient for multivariate continuity can also be seen from

2220-408: The concept of the integral to functions of any number of variables. Double and triple integrals may be used to calculate areas and volumes of regions in the plane and in space. Fubini's theorem guarantees that a multiple integral may be evaluated as a repeated integral or iterated integral as long as the integrand is continuous throughout the domain of integration. The surface integral and

2280-539: The continuity of s ′ ( t ) {\displaystyle s'(t)} , s ′ ( τ ) = s ′ ( t 0 ) + O ( h ) {\displaystyle s'(\tau )=s'(t_{0})+O(h)} as h → 0 {\displaystyle h\to 0} . Substituting these two conditions into 12 , whose limit depends only on s ′ ( t 0 ) {\displaystyle s'(t_{0})} as

2340-417: The conversion between two square units is the square of the conversion between the corresponding length units. the relationship between square feet and square inches is where 144 = 12 = 12 × 12. Similarly: In addition, conversion factors include: There are several other common units for area. The are was the original unit of area in the metric system , with: Though the are has fallen out of use,

2400-520: The countries use SI units as official, many South Asians still use traditional units. Each administrative division has its own area unit, some of them have same names, but with different values. There's no official consensus about the traditional units values. Thus, the conversions between the SI units and the traditional units may have different results, depending on what reference that has been used. Some traditional South Asian units that have fixed value: In

2460-470: The derivative. In vector calculus , the del operator ( ∇ {\displaystyle \nabla } ) is used to define the concepts of gradient , divergence , and curl in terms of partial derivatives. A matrix of partial derivatives, the Jacobian matrix, may be used to represent the derivative of a function between two spaces of arbitrary dimension. The derivative can thus be understood as

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2520-537: The dominant term. It is therefore possible to generate the definition of the directional derivative as follows: The directional derivative of a scalar-valued function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } along the unit vector u ^ {\displaystyle {\hat {\mathbf {u}}}} at some point x 0 ∈ R n {\displaystyle x_{0}\in \mathbb {R} ^{n}}

2580-728: The following example. For example, for a real-valued function f : R 2 → R {\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} } with two real-valued parameters, f ( x , y ) {\displaystyle f(x,y)} , continuity of f {\displaystyle f} in x {\displaystyle x} for fixed y {\displaystyle y} and continuity of f {\displaystyle f} in y {\displaystyle y} for fixed x {\displaystyle x} does not imply continuity of f {\displaystyle f} . Consider It

2640-409: The form of s ( t ) {\displaystyle s(t)} , i.e. the path chosen, not just the point which the limit approaches. For example, consider the function If the point ( 0 , 0 ) {\displaystyle (0,0)} is approached through the line y = k x {\displaystyle y=kx} , or in parametric form: Then the limit along

2700-406: The left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle: However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of

2760-620: The limit and differential along a 1D parametrized curve, reducing the problem to the 1D case. Further higher-dimensional objects can be constructed from these operators. The consequence of the second difference is the existence of multiple types of integration, including line integrals , surface integrals and volume integrals . Due to the non-uniqueness of these integrals, an antiderivative or indefinite integral cannot be properly defined. A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions. A limit along

2820-516: The limit of f {\displaystyle f} to some point x 0 ∈ R n {\displaystyle x_{0}\in \mathbb {R} ^{n}} is L, if and only if for all continuous functions s ( t ) : R → R n {\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}} such that s ( t 0 ) = x 0 {\displaystyle s(t_{0})=x_{0}} . From

2880-403: The notion of the derivative to higher dimensions. A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant. A partial derivative may be thought of as the directional derivative of the function along a coordinate axis. Partial derivatives may be combined in interesting ways to create more complicated expressions of

2940-432: The other hand, if geometry is developed before arithmetic , this formula can be used to define multiplication of real numbers . Most other simple formulas for area follow from the method of dissection . This involves cutting a shape into pieces, whose areas must sum to the area of the original shape. For an example, any parallelogram can be subdivided into a trapezoid and a right triangle , as shown in figure to

3000-421: The parallelogram: Similar arguments can be used to find area formulas for the trapezoid as well as more complicated polygons . The formula for the area of a circle (more properly called the area enclosed by a circle or the area of a disk ) is based on a similar method. Given a circle of radius r , it is possible to partition the circle into sectors , as shown in the figure to the right. Each sector

3060-420: The path as a 1D function f ( s ( t ) ) {\displaystyle f(s(t))} . The limit of f {\displaystyle f} to the point s ( t 0 ) {\displaystyle s(t_{0})} along the path s ( t ) {\displaystyle s(t)} can hence be defined as Note that the value of this limit can be dependent on

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3120-433: The path will be: On the other hand, if the path y = ± x 2 {\displaystyle y=\pm x^{2}} (or parametrically, x ( t ) = t , y ( t ) = ± t 2 {\displaystyle x(t)=t,\,y(t)=\pm t^{2}} ) is chosen, then the limit becomes: Since taking different paths towards the same point yields different values,

3180-593: The precise form of s ( t ) {\displaystyle s(t)} . The derivative of a single-variable function is defined as Using the extension of limits discussed above, one can then extend the definition of the derivative to a scalar-valued function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } along some path s ( t ) : R → R n {\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}} : Unlike limits, for which

3240-1012: The remainder: where τ ∈ [ t 0 , t ] {\displaystyle \tau \in [t_{0},t]} . Substituting this into 10 , where τ ( h ) ∈ [ t 0 , t 0 + h ] {\displaystyle \tau (h)\in [t_{0},t_{0}+h]} . Lipschitz continuity gives us | f ( x ) − f ( y ) | ≤ K | x − y | {\displaystyle |f(x)-f(y)|\leq K|x-y|} for some finite K {\displaystyle K} , ∀ x , y ∈ R n {\displaystyle \forall x,y\in \mathbb {R} ^{n}} . It follows that | f ( x + O ( h ) ) − f ( x ) | ∼ O ( h ) {\displaystyle |f(x+O(h))-f(x)|\sim O(h)} . Note also that given

3300-413: The standard unit of area is the square metre (written as m ), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics , the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number . There are several well-known formulas for

3360-410: The tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book Measurement of a Circle . (The circumference is 2 π r , and the area of a triangle is half the base times the height, yielding the area π r for the disk.) Archimedes approximated

3420-511: The use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists. An approach to defining what is meant by "area" is through axioms . "Area" can be defined as a function from a collection M of a special kinds of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties: It can be proved that such an area function actually exists. Every unit of length has

3480-451: The value depends on the exact form of the path s ( t ) {\displaystyle s(t)} , it can be shown that the derivative along the path depends only on the tangent vector of the path at s ( t 0 ) {\displaystyle s(t_{0})} , i.e. s ′ ( t 0 ) {\displaystyle s'(t_{0})} , provided that f {\displaystyle f}

3540-419: The value of π (and hence the area of a unit-radius circle) with his doubling method , in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular hexagon , then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same with circumscribed polygons ). Heron of Alexandria found what

3600-470: The wild part of Bazaikha is from the villages of Beryozovskiy and Magansk . This Krasnoyarsk Krai location article is a stub . You can help Misplaced Pages by expanding it . This article related to a river in Siberia is a stub . You can help Misplaced Pages by expanding it . Area Area is the measure of a region 's size on a surface . The area of a plane region or plane area refers to

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