Differential geometry - Revision history

DavidB4: new link

2020-09-08T21:21:12Z

new link

DavidB4-bot: /* top */clean up & uniformity

2016-07-13T11:40:48Z

top: clean up & uniformity

Sunda62: This was listen in Relativity, but it's not exactly used there - GR uses pseudo-Riemannian geometry

2010-05-27T14:11:11Z

This was listen in Relativity, but it's not exactly used there - GR uses pseudo-Riemannian geometry

Aschlafly: '''Differential geometry''' is a branch of mathematics which makes use of techniques of analysis, particularly calculus, to study geometric problems.

2009-10-31T03:58:56Z

'''Differential geometry''' is a branch of mathematics which makes use of techniques of analysis, particularly calculus, to study geometric problems.

MarkGall: beefed up article. i've tried to keep it very elementary and with nothing technical, so it's rather fluffy. thoughts?

2009-06-13T15:58:15Z

beefed up article. i've tried to keep it very elementary and with nothing technical, so it's rather fluffy. thoughts?

DRBSouthington: No relation to topic. Italics seemed dubious.

2008-11-11T01:53:56Z

No relation to topic. Italics seemed dubious.

PiperP at 22:56, October 15, 2008

2008-10-15T22:56:12Z

Foxtrot: trying to get red links off geometry

2008-09-28T01:47:31Z

trying to get red links off geometry

New page

'''Differential geometry''' is a [[geometry]] where differences between [[point]]s are measured using a [[derivative]]. Study of differential geometry requires a firm knowledge of [[calculus]] and [[plane geometry]].

[[Category:Geometry]]

@@ Line 6: / Line 6: @@
 * Can the manifold be "flattened out"?  Any piece of spaghetti can be made straight, and yet an orange peel cannot be flattened without tearing.  How can we tell if it is possible to flatten a given surface?  This question leads to the study of ''curvature''.
 * Is there a natural notion of distance on the manifold?  In the case of a surface in R^3, the answer is yes: just define the length of a path in the surface to be the length of that path in R^3!
-* How should one [[derivative|differentiate]] a function?  What does it mean to take the derivative of a function on the sphere?  This question leads to the development of objects called ''connections''.
+* How should one [[Derivative (calculus)|differentiate]] a function?  What does it mean to take the derivative of a function on the sphere?  This question leads to the development of objects called ''connections''.
 Study of differential geometry requires a firm knowledge of [[calculus]] and [[plane geometry]].

@@ Line 1: / Line 1: @@
 '''Differential geometry''' is a branch of mathematics which makes use of techniques of [[analysis]], particularly [[calculus]], to study geometric problems.  Initially, geometers primarily sought to understand the geometry of curves and surfaces in 3-dimensional [[Euclidean space]], and many important early results in the subject are due to [[Carl Friedrich Gauss|Gauss]].  Other early pioneers included [[Bernhard Riemann]] and Tullio Levi-Civita.
-The primary objects of study in differential geometry are ''smooth'' and ''Riemannian'' [[manifold|manifolds]].  A typical example of such an object is a smooth surface in R^3, for example, the unit sphere.  Typical questions that a differential geometer might ask about a manifold include:
+The primary objects of study in differential geometry are ''smooth'' and ''Riemannian'' [[manifold]]s.  A typical example of such an object is a smooth surface in R^3, for example, the unit sphere.  Typical questions that a differential geometer might ask about a manifold include:
-* Given two points, what is the shortest path between those two points staying on the manifold?  These length-minimizing paths are termed ''geodesics''.  On the sphere, the geodesics are [[great circle|great circles]].
+* Given two points, what is the shortest path between those two points staying on the manifold?  These length-minimizing paths are termed ''geodesics''.  On the sphere, the geodesics are [[great circle]]s.
 * Can the manifold be "flattened out"?  Any piece of spaghetti can be made straight, and yet an orange peel cannot be flattened without tearing.  How can we tell if it is possible to flatten a given surface?  This question leads to the study of ''curvature''.
 * Is there a natural notion of distance on the manifold?  In the case of a surface in R^3, the answer is yes: just define the length of a path in the surface to be the length of that path in R^3!

← Older revision		Revision as of 14:11, May 27, 2010
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	Study of differential geometry requires a firm knowledge of [[calculus]] and [[plane geometry]].		Study of differential geometry requires a firm knowledge of [[calculus]] and [[plane geometry]].
		+
		+	A slight modification of differential geometry was the basis of [[General Relativity]]: the modification was replacing ''Riemannian'' manifolds for ''Pseudo-riemannian'' manifolds. In naive terms, in Riemannian manifolds the square of all distances between different points are positive numbers (so the distances can be taken to be the positive square root of their squares), while the latter uses a ''pseudo-metric'' where the square of a distance can be positive, negative or zero.

	[[Category:Geometry]]		[[Category:Geometry]]

← Older revision		Revision as of 03:58, October 31, 2009
Line 1:		Line 1:
−	'''Differential geometry''' is a branch of mathematics which makes use of techniques of [[analysis]] to study geometric problems. Initially, geometers primarily sought to understand the geometry of curves and surfaces in 3-dimensional [[Euclidean space]], and many important early results in the subject are due to [[Carl Friedrich Gauss\|Gauss]]. Other early pioneers included [[Bernhard Riemann]] and Tullio Levi-Civita.	+	'''Differential geometry''' is a branch of mathematics which makes use of techniques of [[analysis]], particularly [[calculus]], to study geometric problems. Initially, geometers primarily sought to understand the geometry of curves and surfaces in 3-dimensional [[Euclidean space]], and many important early results in the subject are due to [[Carl Friedrich Gauss\|Gauss]]. Other early pioneers included [[Bernhard Riemann]] and Tullio Levi-Civita.

	The primary objects of study in differential geometry are ''smooth'' and ''Riemannian'' [[manifold\|manifolds]]. A typical example of such an object is a smooth surface in R^3, for example, the unit sphere. Typical questions that a differential geometer might ask about a manifold include:		The primary objects of study in differential geometry are ''smooth'' and ''Riemannian'' [[manifold\|manifolds]]. A typical example of such an object is a smooth surface in R^3, for example, the unit sphere. Typical questions that a differential geometer might ask about a manifold include:

← Older revision		Revision as of 15:58, June 13, 2009
Line 1:		Line 1:
−	'''Differential geometry''' is a [[geometry]] ~~where differences~~ between [[~~point~~]]~~s are measured using~~ a [[derivative]]. Study of differential geometry requires a firm knowledge of [[calculus]] and [[plane geometry]].	+	'''Differential geometry''' is a branch of mathematics which makes use of techniques of [[analysis]] to study geometric problems. Initially, geometers primarily sought to understand the geometry of curves and surfaces in 3-dimensional [[Euclidean space]], and many important early results in the subject are due to [[Carl Friedrich Gauss\|Gauss]]. Other early pioneers included [[Bernhard Riemann]] and Tullio Levi-Civita.
		+
		+	The primary objects of study in differential geometry are ''smooth'' and ''Riemannian'' [[manifold\|manifolds]]. A typical example of such an object is a smooth surface in R^3, for example, the unit sphere. Typical questions that a differential geometer might ask about a manifold include:
		+
		+	* Given two points, what is the shortest path between those two points staying on the manifold? These length-minimizing paths are termed ''geodesics''. On the sphere, the geodesics are [[great circle\|great circles]].
		+	* Can the manifold be "flattened out"? Any piece of spaghetti can be made straight, and yet an orange peel cannot be flattened without tearing. How can we tell if it is possible to flatten a given surface? This question leads to the study of ''curvature''.
		+	* Is there a natural notion of distance on the manifold? In the case of a surface in R^3, the answer is yes: just define the length of a path in the surface to be the length of that path in R^3!
		+	* How should one [[derivative\|differentiate]] a function? What does it mean to take the derivative of a function on the sphere? This question leads to the development of objects called ''connections''.
		+
		+	Study of differential geometry requires a firm knowledge of [[calculus]] and [[plane geometry]].

	[[Category:Geometry]]		[[Category:Geometry]]

← Older revision		Revision as of 01:53, November 11, 2008
Line 1:		Line 1:
−	'''Differential geometry''' is a [[geometry]] where differences between [[point]]s are measured using a [[derivative]]. Study of differential geometry requires a firm knowledge of [[calculus]] and [[plane geometry]]~~. Please note that number theory is ''just'' a theory~~.	+	'''Differential geometry''' is a [[geometry]] where differences between [[point]]s are measured using a [[derivative]]. Study of differential geometry requires a firm knowledge of [[calculus]] and [[plane geometry]].

	[[Category:Geometry]]		[[Category:Geometry]]

← Older revision		Revision as of 22:56, October 15, 2008
Line 1:		Line 1:
−	'''Differential geometry''' is a [[geometry]] where differences between [[point]]s are measured using a [[derivative]]. Study of differential geometry requires a firm knowledge of [[calculus]] and [[plane geometry]].	+	'''Differential geometry''' is a [[geometry]] where differences between [[point]]s are measured using a [[derivative]]. Study of differential geometry requires a firm knowledge of [[calculus]] and [[plane geometry]]. Please note that number theory is ''just'' a theory.

	[[Category:Geometry]]		[[Category:Geometry]]