Misplaced Pages

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40-460: B − Y {\displaystyle B-Y} indicates a color difference signal between Blue (B) and a Luminance component, as part of a Luminance (Y) and Chrominance (C) color model. It has different meanings depending on the exact model used: R-Y Luminance The procedure for conversion from spectral radiance to luminance is standardized by the CIE and ISO . Brightness

80-506: A + h e i ) − f ( a ) h . {\displaystyle {\begin{aligned}{\frac {\partial }{\partial x_{i}}}f(\mathbf {a} )&=\lim _{h\to 0}{\frac {f(a_{1},\ldots ,a_{i-1},a_{i}+h,a_{i+1}\,\ldots ,a_{n})\ -f(a_{1},\ldots ,a_{i},\dots ,a_{n})}{h}}\\&=\lim _{h\to 0}{\frac {f(\mathbf {a} +h\mathbf {e_{i}} )-f(\mathbf {a} )}{h}}\,.\end{aligned}}} Where e i {\displaystyle \mathbf {e_{i}} }

120-402: A 1 , … , a i − 1 , a i + h , a i + 1 … , a n )   − f ( a 1 , … , a i , … , a n ) h = lim h → 0 f (

160-419: A ) ) . {\displaystyle \nabla f(a)=\left({\frac {\partial f}{\partial x_{1}}}(a),\ldots ,{\frac {\partial f}{\partial x_{n}}}(a)\right).} This vector is called the gradient of f at a . If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇ f which takes the point a to the vector ∇ f ( a ) . Consequently, the gradient produces

200-392: A scalar function f ( x ) = f ( x 1 , x 2 , … , x n ) {\displaystyle f(\mathbf {x} )=f(x_{1},x_{2},\ldots ,x_{n})} along a vector v = ( v 1 , … , v n ) {\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n})}

240-2472: A vector field . A common abuse of notation is to define the del operator ( ∇ ) as follows in three-dimensional Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} with unit vectors i ^ , j ^ , k ^ {\displaystyle {\hat {\mathbf {i} }},{\hat {\mathbf {j} }},{\hat {\mathbf {k} }}} : ∇ = [ ∂ ∂ x ] i ^ + [ ∂ ∂ y ] j ^ + [ ∂ ∂ z ] k ^ {\displaystyle \nabla =\left[{\frac {\partial }{\partial x}}\right]{\hat {\mathbf {i} }}+\left[{\frac {\partial }{\partial y}}\right]{\hat {\mathbf {j} }}+\left[{\frac {\partial }{\partial z}}\right]{\hat {\mathbf {k} }}} Or, more generally, for n -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} with coordinates x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} and unit vectors e ^ 1 , … , e ^ n {\displaystyle {\hat {\mathbf {e} }}_{1},\ldots ,{\hat {\mathbf {e} }}_{n}} : ∇ = ∑ j = 1 n [ ∂ ∂ x j ] e ^ j = [ ∂ ∂ x 1 ] e ^ 1 + [ ∂ ∂ x 2 ] e ^ 2 + ⋯ + [ ∂ ∂ x n ] e ^ n {\displaystyle \nabla =\sum _{j=1}^{n}\left[{\frac {\partial }{\partial x_{j}}}\right]{\hat {\mathbf {e} }}_{j}=\left[{\frac {\partial }{\partial x_{1}}}\right]{\hat {\mathbf {e} }}_{1}+\left[{\frac {\partial }{\partial x_{2}}}\right]{\hat {\mathbf {e} }}_{2}+\dots +\left[{\frac {\partial }{\partial x_{n}}}\right]{\hat {\mathbf {e} }}_{n}} The directional derivative of

280-671: A case, evaluation of the function must be expressed in an unwieldy manner as ∂ f ( x , y , z ) ∂ x ( 17 , u + v , v 2 ) {\displaystyle {\frac {\partial f(x,y,z)}{\partial x}}(17,u+v,v^{2})} or ∂ f ( x , y , z ) ∂ x | ( x , y , z ) = ( 17 , u + v , v 2 ) {\displaystyle \left.{\frac {\partial f(x,y,z)}{\partial x}}\right|_{(x,y,z)=(17,u+v,v^{2})}} in order to use

320-568: A function of several variables is the case of a scalar-valued function f ( x 1 , … , x n ) {\displaystyle f(x_{1},\ldots ,x_{n})} on a domain in Euclidean space R n {\displaystyle \mathbb {R} ^{n}} (e.g., on R 2 {\displaystyle \mathbb {R} ^{2}} or R 3 {\displaystyle \mathbb {R} ^{3}} ). In this case f has

360-509: A function. The partial derivative of f at the point a = ( a 1 , … , a n ) ∈ U {\displaystyle \mathbf {a} =(a_{1},\ldots ,a_{n})\in U} with respect to the i -th variable x i is defined as ∂ ∂ x i f ( a ) = lim h → 0 f (

400-520: A lossless medium, the luminance does not change along a given light ray . As the ray crosses an arbitrary surface S , the luminance is given by L v = d 2 Φ v d S d Ω S cos ⁡ θ S {\displaystyle L_{\mathrm {v} }={\frac {\mathrm {d} ^{2}\Phi _{\mathrm {v} }}{\mathrm {d} S\,\mathrm {d} \Omega _{S}\cos \theta _{S}}}} where More generally,

440-479: A partial derivative ∂ f / ∂ x j {\displaystyle \partial f/\partial x_{j}} with respect to each variable x j . At the point a , these partial derivatives define the vector ∇ f ( a ) = ( ∂ f ∂ x 1 ( a ) , … , ∂ f ∂ x n (

SECTION 10

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480-502: A partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in: f x ′ ( x , y , … ) , ∂ f ∂ x ( x , y , … ) . {\displaystyle f'_{x}(x,y,\ldots ),{\frac {\partial f}{\partial x}}(x,y,\ldots ).} The symbol used to denote partial derivatives

520-618: A way similar to the way a digital camera records color images. The luminance of a specified point of a light source, in a specified direction, is defined by the mixed partial derivative L v = d 2 Φ v d Σ d Ω Σ cos ⁡ θ Σ {\displaystyle L_{\mathrm {v} }={\frac {\mathrm {d} ^{2}\Phi _{\mathrm {v} }}{\mathrm {d} \Sigma \,\mathrm {d} \Omega _{\Sigma }\cos \theta _{\Sigma }}}} where If light travels through

560-603: Is totally differentiable in that neighborhood and the total derivative is continuous. In this case, it is said that f is a C function. This can be used to generalize for vector valued functions, f : U → R m {\displaystyle f:U\to \mathbb {R} ^{m}} , by carefully using a componentwise argument. The partial derivative ∂ f ∂ x {\textstyle {\frac {\partial f}{\partial x}}} can be seen as another function defined on U and can again be partially differentiated. If

600-602: Is ∂ . One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences . The modern partial derivative notation was created by Adrien-Marie Legendre (1786), although he later abandoned it; Carl Gustav Jacob Jacobi reintroduced the symbol in 1841. Like ordinary derivatives, the partial derivative is defined as a limit . Let U be an open subset of R n {\displaystyle \mathbb {R} ^{n}} and f : U → R {\displaystyle f:U\to \mathbb {R} }

640-460: Is a function of more than one variable. For instance, z = f ( x , y ) = x 2 + x y + y 2 . {\displaystyle z=f(x,y)=x^{2}+xy+y^{2}.} The graph of this function defines a surface in Euclidean space . To every point on this surface, there are an infinite number of tangent lines . Partial differentiation

680-518: Is the function ∇ v f {\displaystyle \nabla _{\mathbf {v} }{f}} defined by the limit ∇ v f ( x ) = lim h → 0 f ( x + h v ) − f ( x ) h . {\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\to 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h}}.} Suppose that f

720-424: Is the stilb , which is equal to one candela per square centimetre or 10 kcd/m . Luminance is often used to characterize emission or reflection from flat, diffuse surfaces. Luminance levels indicate how much luminous power could be detected by the human eye looking at a particular surface from a particular angle of view . Luminance is thus an indicator of how bright the surface will appear. In this case,

760-517: Is the term for the subjective impression of the objective luminance measurement standard (see Objectivity (science) § Objectivity in measurement for the importance of this contrast). The SI unit for luminance is candela per square metre (cd/m ). A non-SI term for the same unit is the nit . The unit in the Centimetre–gram–second system of units (CGS) (which predated the SI system)

800-402: Is the unit vector of i -th variable x i . Even if all partial derivatives ∂ f / ∂ x i ( a ) {\displaystyle \partial f/\partial x_{i}(a)} exist at a given point a , the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of a and are continuous there, then f

840-434: Is the act of choosing one of these lines and finding its slope . Usually, the lines of most interest are those that are parallel to the xz -plane, and those that are parallel to the yz -plane (which result from holding either y or x constant, respectively). To find the slope of the line tangent to the function at P (1, 1) and parallel to the xz -plane, we treat y as a constant. The graph and this plane are shown on

SECTION 20

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880-412: The total derivative , in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry . The partial derivative of a function f ( x , y , … ) {\displaystyle f(x,y,\dots )} with respect to the variable x {\displaystyle x} is variously denoted by It can be thought of as

920-533: The Leibniz notation. Thus, in these cases, it may be preferable to use the Euler differential operator notation with D i {\displaystyle D_{i}} as the partial derivative symbol with respect to the i -th variable. For instance, one would write D 1 f ( 17 , u + v , v 2 ) {\displaystyle D_{1}f(17,u+v,v^{2})} for

960-422: The candela per square metre. Luminance is essentially the same as surface brightness , the term used in astronomy. This is measured with a logarithmic scale, magnitudes per square arcsecond (MPSAS). Mixed partial derivative In mathematics , a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to

1000-710: The direction of derivative is not repeated, it is called a mixed partial derivative . If all mixed second order partial derivatives are continuous at a point (or on a set), f is termed a C function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem : ∂ 2 f ∂ x i ∂ x j = ∂ 2 f ∂ x j ∂ x i . {\displaystyle {\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}={\frac {\partial ^{2}f}{\partial x_{j}\partial x_{i}}}.} For

1040-711: The example described above, while the expression D 1 f {\displaystyle D_{1}f} represents the partial derivative function with respect to the first variable. For higher order partial derivatives, the partial derivative (function) of D i f {\displaystyle D_{i}f} with respect to the j -th variable is denoted D j ( D i f ) = D i , j f {\displaystyle D_{j}(D_{i}f)=D_{i,j}f} . That is, D j ∘ D i = D i , j {\displaystyle D_{j}\circ D_{i}=D_{i,j}} , so that

1080-462: The eye to lasers, which are high luminance sources. The IEC 62471 series gives guidance for evaluating the photobiological safety of lamps and lamp systems including luminaires. Specifically it specifies the exposure limits, reference measurement technique and classification scheme for the evaluation and control of photobiological hazards from all electrically powered incoherent broadband sources of optical radiation, including LEDs but excluding lasers, in

1120-2203: The following examples, let f be a function in x , y , and z . First-order partial derivatives: ∂ f ∂ x = f x ′ = ∂ x f . {\displaystyle {\frac {\partial f}{\partial x}}=f'_{x}=\partial _{x}f.} Second-order partial derivatives: ∂ 2 f ∂ x 2 = f x x ″ = ∂ x x f = ∂ x 2 f . {\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}=f''_{xx}=\partial _{xx}f=\partial _{x}^{2}f.} Second-order mixed derivatives : ∂ 2 f ∂ y ∂ x = ∂ ∂ y ( ∂ f ∂ x ) = ( f x ′ ) y ′ = f x y ″ = ∂ y x f = ∂ y ∂ x f . {\displaystyle {\frac {\partial ^{2}f}{\partial y\,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial f}{\partial x}}\right)=(f'_{x})'_{y}=f''_{xy}=\partial _{yx}f=\partial _{y}\partial _{x}f.} Higher-order partial and mixed derivatives: ∂ i + j + k f ∂ x i ∂ y j ∂ z k = f ( i , j , k ) = ∂ x i ∂ y j ∂ z k f . {\displaystyle {\frac {\partial ^{i+j+k}f}{\partial x^{i}\partial y^{j}\partial z^{k}}}=f^{(i,j,k)}=\partial _{x}^{i}\partial _{y}^{j}\partial _{z}^{k}f.} When dealing with functions of multiple variables, some of these variables may be related to each other, thus it may be necessary to specify explicitly which variables are being held constant to avoid ambiguity. In fields such as statistical mechanics ,

1160-566: The function at a specific point are conflated by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an expression like ∂ f ( x , y , z ) ∂ x {\displaystyle {\frac {\partial f(x,y,z)}{\partial x}}} is used for the function, while ∂ f ( u , v , w ) ∂ u {\displaystyle {\frac {\partial f(u,v,w)}{\partial u}}} might be used for

1200-454: The integral covers all the directions of emission Ω Σ , In the case of a perfectly diffuse reflector (also called a Lambertian reflector ), the luminance is isotropic, per Lambert's cosine law . Then the relationship is simply L v = E v R π . {\displaystyle L_{\text{v}}={\frac {E_{\text{v}}R}{\pi }}.} A variety of units have been used for luminance, besides

1240-759: The luminance along a light ray can be defined as L v = n 2 d Φ v d G {\displaystyle L_{\mathrm {v} }=n^{2}{\frac {\mathrm {d} \Phi _{\mathrm {v} }}{\mathrm {d} G}}} where The luminance of a reflecting surface is related to the illuminance it receives: ∫ Ω Σ L v d Ω Σ cos ⁡ θ Σ = M v = E v R , {\displaystyle \int _{\Omega _{\Sigma }}L_{\text{v}}\mathrm {d} \Omega _{\Sigma }\cos \theta _{\Sigma }=M_{\text{v}}=E_{\text{v}}R,} where

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1280-406: The luminance comes out to be the same assuming there is no loss at the lens. The image can never be "brighter" than the source. Retinal damage can occur when the eye is exposed to high luminance. Damage can occur because of local heating of the retina. Photochemical effects can also cause damage, especially at short wavelengths. The IEC 60825 series gives guidance on safety relating to exposure of

1320-410: The output is the same as the input luminance. For real, passive optical systems, the output luminance is at most equal to the input. As an example, if one uses a lens to form an image that is smaller than the source object, the luminous power is concentrated into a smaller area, meaning that the illuminance is higher at the image. The light at the image plane, however, fills a larger solid angle so

1360-401: The partial derivative of f with respect to x , holding y and z constant, is often expressed as ( ∂ f ∂ x ) y , z . {\displaystyle \left({\frac {\partial f}{\partial x}}\right)_{y,z}.} Conventionally, for clarity and simplicity of notation, the partial derivative function and the value of

1400-520: The rate of change of the function in the x {\displaystyle x} -direction. Sometimes, for z = f ( x , y , … ) {\displaystyle z=f(x,y,\ldots )} , the partial derivative of z {\displaystyle z} with respect to x {\displaystyle x} is denoted as ∂ z ∂ x . {\displaystyle {\tfrac {\partial z}{\partial x}}.} Since

1440-411: The right. Below, we see how the function looks on the plane y = 1 . By finding the derivative of the equation while assuming that y is a constant, we find that the slope of f at the point ( x , y ) is: ∂ z ∂ x = 2 x + y . {\displaystyle {\frac {\partial z}{\partial x}}=2x+y.} So at (1, 1) , by substitution,

1480-409: The solid angle of interest is the solid angle subtended by the eye's pupil . Luminance is used in the video industry to characterize the brightness of displays. A typical computer display emits between 50 and 300 cd/m . The sun has a luminance of about 1.6 × 10  cd/m at noon. Luminance is invariant in geometric optics . This means that for an ideal optical system, the luminance at

1520-442: The value of the function at the point ( x , y , z ) = ( u , v , w ) {\displaystyle (x,y,z)=(u,v,w)} . However, this convention breaks down when we want to evaluate the partial derivative at a point like ( x , y , z ) = ( 17 , u + v , v 2 ) {\displaystyle (x,y,z)=(17,u+v,v^{2})} . In such

1560-417: The variables are listed in the order in which the derivatives are taken, and thus, in reverse order of how the composition of operators is usually notated. Of course, Clairaut's theorem implies that D i , j = D j , i {\displaystyle D_{i,j}=D_{j,i}} as long as comparatively mild regularity conditions on f are satisfied. An important example of

1600-509: The wavelength range from 200 nm through 3000 nm . This standard was prepared as Standard CIE S 009:2002 by the International Commission on Illumination. A luminance meter is a device used in photometry that can measure the luminance in a particular direction and with a particular solid angle . The simplest devices measure the luminance in a single direction while imaging luminance meters measure luminance in

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