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Topic: Directional derivative

Related Topics

Partial derivative

Derivative

Gradient

Lie derivative

Derivation (abstract algebra)

Limit (mathematics)

Product rule

Quotient rule

Scalar field

Dot product

Critical point

	PlanetMath: derivative
		Qualitatively the derivative is a measure of the change of a function in a small region around a specified point.
		Since the notion of a manifold was constructed specifically to generalize the notion of a derivative, this seems like the end of the road for this entry.
		This is version 24 of derivative, born on 2002-05-31, modified 2006-12-08.
	planetmath.org /encyclopedia/DirectionalDerivative2.html (801 words)

	Evaluating the performance of the derivative measures
		Recall that the masks derived in Appendix B are not themselves the first and second directional derivative measures needed for histogram volume calculation.
		The directional derivative measures rely on the masks to provide partial derivative measurement, and the partial derivatives are composed into the final directional derivative measure according to the formulas in Section 4.3.
		Unfortunately, due to the complexity of the directional derivative formulas in Equations C.2, C.3, and C.5, the task is outside the scope of this thesis.
	www.cs.utah.edu /~gk/MS/html/node34.html (1416 words)

	[No title]
		This curve is the intersection of the plane x=x0 with the surface z=f(x,y).
		It is the directional derivative in the direction of the unit vector "j".
		The directional derivative of f in the direction of u=(a,b) is the projection of the gradient of f onto the direction u.
	omega.albany.edu:8008 /mat214dir/directional-derivatives-dir/test (733 words)

	Directional/Partial Derivatives
		The directional derivative along the vector v is the limit of f(x+hv)-f(x) over h as h approaches 0.
		If you are walking east, the directional derivative is called the partial of f with respect to x, and is written f∂x.
		Like the second derivative on a graph, the second partial tells us how the slope, in the x direction, is changing as we move east.
	www.mathreference.com /ca-mv,part.html (726 words)

	Directional derivative - Wikipedia, the free encyclopedia
		In mathematics, the directional derivative of a multivariate differentiable function along a given vector intuitively represents the rate of change of the function in the direction of that vector.
		It therefore generalizes the notion of a partial derivative, in which the direction is always taken parallel to one of the coordinate axes.
		A normal derivative is a directional derivative taken in the direction normal (that is, orthogonal) to some surface in space, or more generally along a normal vector field orthogonal to some hypersurface.
	en.wikipedia.org /wiki/Directional_derivative (235 words)

	Derivative - Real Time & Delayed Quotes, Charts, News and Data for Futures, Stocks, Commodities and Indexes - ...
		The derivative of a function f at x is geometrically the slope of the tangent line to the graph of f at x.
		In this case, the Second Derivative Test can still be used to characterize critical points, by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point.
		The common thread is that the derivative at a point serves as a linear approximation of the function at that point.
	www.tradesignals.com /glossary/Derivative (2152 words)

	GRDGRADIENT
		grdgradient may be used to compute the directional derivative in a given direction (−A), or the direction (−S) [and the magnitude (−D)] of the vector gradient of the data.
		Azimuthal direction for a directional derivative; azim is the angle in the x,y plane measured in degrees positive clockwise from north (the +y direction) toward east (the +x direction).
		The negative of the directional derivative, −[dz/dxsin(azim) + dz/dycos(azim)], is found; negation yields positive values when the slope of z(x,y) is downhill in the azim direction, the correct sense for shading the illumination of an image (see grdimage and grdview) by a light source above the x,y plane shining from the azim direction.
	gmt.soest.hawaii.edu /gmt/doc/html/grdgradient.html (871 words)

	13.6
		Below is a contour plot with the gradient vector at (-1,0) pointing in the direction of steepest ascent 2i + j.
		The relationship of the partial derivatives of f to the directional derivatives are f
		In the direction of i + j, we have:
	www.ac.cc.md.us /~donr/CalcIII/unit3/lesson6/u3l6.html (867 words)

	GRDGRADIENT
		grdgradient may be used to compute the directional derivative in a given direction (−A), or the direction (−S) [and the magnitude (−D)] of the vector gradient of the data.
		Azimuthal direction for a directional derivative; azim is the angle in the x,y plane measured in degrees positive clockwise from north (the +y direction) toward east (the +x direction).
		The negative of the directional derivative, −[dz/dxsin(azim) + dz/dycos(azim)], is found; negation yields positive values when the slope of z(x,y) is downhill in the azim direction, the correct sense for shading the illumination of an image (see grdimage and grdview) by a light source above the x,y plane shining from the azim direction.
	www.soest.hawaii.edu /gmt/gmt/doc/html/grdgradient.html (871 words)

	Directional Derivative and the Gradient
		The "direction" in the directional derivative relates to a direction vector for the line.
		Thus the directional derivative will be maximized when the measure of the angle between the gradient and the direction vector is zero.
		The horizontal vectors correspond to the direction vectors
	www2.scc-fl.edu /lvosbury/CalculusIII_Folder/DirectionalDerivative.htm (832 words)

	Directional Derivative Example (Site not responding. Last check: 2007-10-30)
		Suppose the wire mesh represents the surface of a mountain and a nearby valley.
		That's the red arrow, where the direction of the arrow shows you which direction the climber is facing.
		If the arrow is pointing up, then the directional derivative in that direction is positive.
	www.math.umn.edu /~rogness/multivar/dirderiv.shtml (245 words)

	PlanetMath: directional derivative
		See Also: partial derivative, derivative, derivative notation, Jacobian matrix, gradient, fixed points of normal functions, Hessian matrix
		derivative with respect to a vector, partial derivative with respect to a vector
		This is version 11 of directional derivative, born on 2001-11-14, modified 2005-04-16.
	planetmath.org /encyclopedia/PartialDerivativeWithRespectToAVector.html (104 words)

	6.6 The Gradient and Directional Derivatives
		The gradient vector is in the direction of the projection of the normal to the tangent hyper-plane into the hyper-plane of coordinates.
		The component of the gradient vector in the direction of any axis is the partial derivative of f with respect to the corresponding distance variable in that direction.
		The slope of that linear approximation is the directional derivative of the field at that edge, in the direction of the cutting half plane.
	www-math.mit.edu /18.013A/HTML/chapter06/section06.html (541 words)

	The Directional Derivatives
		derivative of f with respect to that coordinate as the other coordinates are constant.
		The derivative of f with respect to s is not a partial derivative because as s changes, all of the coordinates usually
		On the next page we shall derive another equation for the directional derivative that is much easier to remember.
	hemsidor.torget.se /users/m/mauritz/math/field/dirder.htm (454 words)

	[BUG] A geometric approach to gradients
		DEFINE the gradient of a function to be the vector whose direction is that in which f increases the fastest, and whose magnitude is the rate of increase in that direction.
		I chose to save the concept of directional derivative for a separate lecture, reiterating that the number of contour lines crossed, and hence the rate of change, in any direction is the dot product with "Mg", in this case the gradient.
		It was also not until this later lecture that I expressed directional derivatives in terms of limits, arguing by analogy with partial derivatives, but not actually deriving the formula normally used to define the directional derivative.
	www.math.oregonstate.edu /bridge/bug/2005/000075.html (640 words)

	One dimensional masks from reconstruction filters
		Figure 4.3 demonstrated the use of a simple mask which was used to measure the first derivative in one dimensional data.
		In this case, the second derivative of the reconstruction kernel must exist at all integer locations for the mask to be valid.
		In this section we return to the strict usage; ``first derivative'' means the first derivative of a function of one variable (and likewise for a second derivative).
	www.cs.utah.edu /~gk/MS/html/node31.html (518 words)

	Covariant derivative - Wikipedia, the free encyclopedia
		Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach via a connection form.
		The covariant derivative is a generalization of the directional derivative from vector calculus.
		The covariant derivative can be described by a "tensor" in a fixed coordinate chart, but it is not a true tensor in the sense that it is not invariant under coordinate changes.
	en.wikipedia.org /wiki/Covariant_derivative (2400 words)

	MultiVariate Directional Derivative \| MaplePrimes
		The directional derivative will evaluate to a simple numerical answer if the derivatives of the function are constants.
		The directional derivative can be evaluated at any given point by substituting the value of that point's coordinates in the derivative.
		Once the directional derivative is defined, entering a vector, coordinates or other parameters requires that you call the directional derivative function again and presumably update it.
	beta.mapleprimes.com /forum/multivariate-directional-derivative-0 (463 words)

	Structure Tensor - Tutorial and Demonstration of the uses of 'structure tensors' in gradient representation
		An example is shown against a horizontal step-edge image with the directional derivative overlaid with a red-arrow.
		To reflect this periodicity, the directional derivative, when applied to the extraction of gradient-based structures, should also have an arrow pointed in the opposite direction.
		Again, the directional derivative magnitude is zero as there is no gradient information with which to calculate.
	www.mathworks.com /matlabcentral/files/12362/content/ST_pub/html/structureTensorDemo.html (2762 words)

	Edge Detection
		Of particular interest in edge detection are two functions that can be expressed in terms of these directional derivatives: the gradient magnitude and the gradient orientation.
		Note that each of the Sobel edge masks is a combination of a digital differentiator in one of the spatial directions and a smoothing operator in the other.
		The edges are identified by the location of zero crossings (recall that the second derivative changes sign in the vicinity of maxima of the first derivative).
	library.wolfram.com /examples/edgedetection (986 words)

	What is a Directional Derivative?
		The partial derivative fy(x0,y0) is a special case of a directional derivative.
		the partial fy is the rate of change of f when r moves a bit from r in the direction of j.
		so the directional derivative of f in the direction of u, "Duf" is the rate of change of f when we move from r a little bit in the direction of u.
	omega.albany.edu:8008 /calc3/directional-derivatives-dir/define-m2h.html (725 words)

	calculus 3 and 4
		Multivariable Calculus (calc 3) problem - directional derivatives - Consider the function f(x,y,z) = (e^z)ln(x^2 + y^2) a) Is there a vector r such that the directional derivative of f at (1,1,0) in the direction of r equals 1?
		Find the maximum directional derivative of f at P and the direction in which it occurs:...
		Directional derivative - (See attached file for full problem description with proper equations) --- Find the directional derivative of the function f(x,y)=3x2-y2+5xy at the point P(2,-1) in the direction of a=(-3i-4j).
	www.brainmass.com /homework-help/math/other/13784 (268 words)

	4 (Site not responding. Last check: 2007-10-30)
		Defined partial derivatives, as directional derivatives along the axis directions, with orthogonal variables held constant.
		Specifically, wrote down expressions for the differential of a function, and for its directional derivatives in arbitrary direction.
		Noted this unspoken assumption in partial derivatives, of which path the function varies along, holding which variables constant; and the need to be very careful in taking partial derivatives to respect that assumption, by avoiding carelessly mixed coordinate systems.
	www.emory.edu /PHYSICS/Faculty/Benson/320/notes/4/4.html (281 words)

	World Web Math: Vector Calculus: Gradients
		Say we move away from point P in a specified direction that is not necessarily along one of the three axes.
		But the gradient vector still points in the direction of greatest increase of the function and any vector perpendicular to the gradient will have a zero directional derivative.
		And this is the value of the direction derivative in the directio of vector v.
	web.mit.edu /wwmath/vectorc/scalar/grad.html (719 words)

	Gravity: Kodaira's Theorem
		The ordinary derivative of f in the direction of coordinate number i.
		The covariant derivative of f in direction i.
		The ordinary derivative of r in direction i of the first argument, or direction j of the second.
	www.math.ucla.edu /~jimc/klein_h/kodaira.html (1043 words)

	What is Automatic Differentiation ?
		For that, the derivative object that must be computed along with each value v is the vector of all partial derivatives of v with respect to each input.
		In other words, this requires one computation of the directional derivative per input variable, and even less than that, if one knows that the Jacobian matrix is sufficiently sparse.
		Like with the directional derivatives, one can compute the whole Jacobian matrix by repeatedly computing gradients, for each canonical direction in the output space, or fewer than that when the Jacobian is sparse.
	pauillac.inria.fr /cdrom/www/tapenade/ad/whatisad.html (1817 words)

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