- published: 10 Jun 2018
- views: 160446
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to defining integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.
For instance, the expression f(x) dx from one-variable calculus is called a 1-form, and can be integrated over an interval [a, b] in the domain of f:
and similarly the expression f(x, y, z) dx ∧ dy + g(x, y, z) dx ∧ dz + h(x, y, z) dy ∧ dz is a 2-form that has a surface integral over an oriented surface S:
Likewise, a 3-form f(x, y, z) dx ∧ dy ∧ dz represents a volume element that can be integrated over a region of space.
The algebra of differential forms is organized in a way that naturally reflects the orientation of the domain of integration. There is an operation d on differential forms known as the exterior derivative that, when acting on a k-form, produces a (k + 1)-form. This operation extends the differential of a function, and the divergence and the curl of a vector field in an appropriate sense that makes the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem special cases of the same general result, known in this context also as the general Stokes' theorem. In a deeper way, this theorem relates the topology of the domain of integration to the structure of the differential forms themselves; the precise connection is known as de Rham's theorem.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Differential_form
Discrete differential geometry
Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics and topological combinatorics.
See also
References
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Discrete_differential_geometry
Differential geometry
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.
Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques characteristic of this field.
History of Development
Differential geometry arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces. Mathematical analysis of curves and surfaces had been developed to answer some of the nagging and unanswered questions that appeared in Calculus, like the reasons for relationships between complex shapes and curves, series and analytic functions. These unanswered questions indicated greater, hidden relationships and symmetries in nature, which the standard methods of analysis could not address.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Differential_geometry
Differential
Differential may refer to:
Mathematics
- Differential of a function, represents a change in the linearization of a function
- Total differential is its generalization for functions of multiple variables
- Differential (infinitesimal) (e.g. dx, dy, dt etc.) are interpreted as infinitesimals
- Differential topology, in multivariable calculus, the differential of a smooth map between Euclidean spaces or differentiable manifolds is the approximating linear map between the tangent spaces, called pushforward (differential)
- Total differential is its generalization for functions of multiple variables
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Differential
Differential (infinitesimal)
The term differential is used in calculus to refer to an infinitesimal (infinitely small) change in some varying quantity. For example, if x is a variable, then a change in the value of x is often denoted Δx (pronounced delta x). The differential dx represents an infinitely small change in the variable x. The idea of an infinitely small or infinitely slow change is extremely useful intuitively, and there are a number of ways to make the notion mathematically precise.
Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using derivatives. If y is a function of x, then the differential dy of y is related to dx by the formula
where dy/dx denotes the derivative of y with respect to x. This formula summarizes the intuitive idea that the derivative of y with respect to x is the limit of the ratio of differences Δy/Δx as Δx becomes infinitesimal.
There are several approaches for making the notion of differentials mathematically precise.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Differential_(infinitesimal)
Differential (mechanical device)
A differential is a particular type of simple planetary gear train that has the property that the angular velocity of its carrier is the average of the angular velocities of its sun and annular gears. This is accomplished by packaging the gear train so it has a fixed carrier train ratio R = -1, which means the gears corresponding to the sun and annular gears are the same size. This can be done by engaging the planet gears of two identical and coaxial epicyclic gear trains to form a spur gear differential. Another approach is to use bevel gears for the sun and annular gears and a bevel gear as the planet, which is known as a bevel gear differential.
Overview
In automobiles and other wheeled vehicles, the differential allows the outer drive wheel to rotate faster than the inner drive wheel during a turn. This is necessary when the vehicle turns, making the wheel that is traveling around the outside of the turning curve roll farther and faster than the other. The average of the rotational speed of the two driving wheels equals the input rotational speed of the drive shaft. An increase in the speed of one wheel is balanced by a decrease in the speed of the other.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Differential_(mechanical_device)
Geometry
Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales (6th century BC). By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. In the classical world, both geometry and astronomy were considered to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Geometry
Podcasts:
-
Introduction to Differential Geometry: Curves
In this video, I introduce Differential Geometry by talking about curves. Curves and surfaces are the two foundational structures for differential geometry, which is why I'm introducing this series by defining curves. After defining level curves, parametrized curves, and tangent vectors, I solve a short example where I convert a level curve to a parametrized curve and then find its tangent vector. Questions/requests? Let me know in the comments! Pre-requisites: A background in Multivariable Calculus (Calculus 3) is helpful, but even if you know the material until Calculus 2, you probably still won't be lost. Lecture Notes: https://drive.google.com/open?id=1CirfXRYfjS8eKB7TVwEWkAT-8nTzpFEQ Patreon: https://www.patreon.com/user?u=4354534 Twitter: https://twitter.com/FacultyOfKhan Specia...
published: 10 Jun 2018 -
Differential Geometry - Claudio Arezzo - Lecture 01
published: 17 Nov 2016 -
Differential Geometry | Math History | NJ Wildberger
Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the work of C. Huygens on involutes and evolutes, and the related notions of curvature and osculating circle. We discuss involutes of the catenary (yielding the tractrix), cycloid and parabola. The evolute of the parabola is a semi-cubical parabola. For space curves we describe the tangent line, osculating plane, principle normal and binormal. Surfaces were studied by Euler, who investigated curvatures of planar sections and by Gauss, who realized that the product of Euler's two principal curvatures gave a new notion of curvature intrinsic to a surface. Curvature was ultimately extended by Riemann to higher dimensions, and plays today a...
published: 07 May 2012 -
Lecture 2B: Introduction to Manifolds (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
published: 06 Feb 2021 -
Lecture 1: Overview (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
published: 04 Feb 2021 -
Curvature: Intuition and Derivation | Differential Geometry
In my 5th video on #DifferentialGeometry, I define the #Curvature for both a unit speed curve reparametrized with respect to arc length and a regular curve parameterized by t. I describe the intuition behind the curvature as the extent to which a curve deviates from a straight line (zero curvature). I then derive the expression for curvature for both a unit speed curve and a regular curve, using the #UnitNormal. Questions/feedback? Let me know in the comments! Pre-reqs: the previous videos in my playlist https://www.youtube.com/playlist?list=PLdgVBOaXkb9DJjk8V0-RkXTnD4ZXUOFsc Lecture Notes: https://drive.google.com/open?id=1_40zI8E2r81zOmb_nkhWbN7q2gSqUc7x Patreon: https://www.patreon.com/user?u=4354534 Twitter: https://twitter.com/FacultyOfKhan Special Thanks to my Patrons: Cesar Garz...
published: 25 Mar 2020 -
Lecture 5: Differential Forms (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
published: 12 Feb 2021
developed with YouTube
10:25
Introduction to Differential Geometry: Curves
- Order: Reorder
- Duration: 10:25
- Uploaded Date: 10 Jun 2018
- views: 160446
In this video, I introduce Differential Geometry by talking about curves. Curves and surfaces are the two foundational structures for differential geometry, whi...
In this video, I introduce Differential Geometry by talking about curves. Curves and surfaces are the two foundational structures for differential geometry, which is why I'm introducing this series by defining curves.
After defining level curves, parametrized curves, and tangent vectors, I solve a short example where I convert a level curve to a parametrized curve and then find its tangent vector.
Questions/requests? Let me know in the comments!
Pre-requisites: A background in Multivariable Calculus (Calculus 3) is helpful, but even if you know the material until Calculus 2, you probably still won't be lost.
Lecture Notes: https://drive.google.com/open?id=1CirfXRYfjS8eKB7TVwEWkAT-8nTzpFEQ
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- Yuan Gao
- James Mark Wilson
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
https://wn.com/Introduction_To_Differential_Geometry_Curves
In this video, I introduce Differential Geometry by talking about curves. Curves and surfaces are the two foundational structures for differential geometry, which is why I'm introducing this series by defining curves.
After defining level curves, parametrized curves, and tangent vectors, I solve a short example where I convert a level curve to a parametrized curve and then find its tangent vector.
Questions/requests? Let me know in the comments!
Pre-requisites: A background in Multivariable Calculus (Calculus 3) is helpful, but even if you know the material until Calculus 2, you probably still won't be lost.
Lecture Notes: https://drive.google.com/open?id=1CirfXRYfjS8eKB7TVwEWkAT-8nTzpFEQ
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- Yuan Gao
- James Mark Wilson
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
1:29:40
Differential Geometry - Claudio Arezzo - Lecture 01
- Order: Reorder
- Duration: 1:29:40
- Uploaded Date: 17 Nov 2016
- views: 159616
- published: 17 Nov 2016
- views: 159616
51:32
Differential Geometry | Math History | NJ Wildberger
- Order: Reorder
- Duration: 51:32
- Uploaded Date: 07 May 2012
- views: 167147
Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the w...
Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the work of C. Huygens on involutes and evolutes, and the related notions of curvature and osculating circle. We discuss involutes of the catenary (yielding the tractrix), cycloid and parabola. The evolute of the parabola is a semi-cubical parabola. For space curves we describe the tangent line, osculating plane, principle normal and binormal.
Surfaces were studied by Euler, who investigated curvatures of planar sections and by Gauss, who realized that the product of Euler's two principal curvatures gave a new notion of curvature intrinsic to a surface. Curvature was ultimately extended by Riemann to higher dimensions, and plays today a major role in modern physics, due to the work of Einstein.
If you like this topic, and want to learn more, make sure you don't miss Wildberger's exciting new course on Differential Geometry! See the Playlist DiffGeom, at this channel.
************************
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger
My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated
***********************
Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: https://www.youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************
And here are the Wild Egg Maths Playlists:
Triangle Centres: https://www.youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
м
https://wn.com/Differential_Geometry_|_Math_History_|_Nj_Wildberger
Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the work of C. Huygens on involutes and evolutes, and the related notions of curvature and osculating circle. We discuss involutes of the catenary (yielding the tractrix), cycloid and parabola. The evolute of the parabola is a semi-cubical parabola. For space curves we describe the tangent line, osculating plane, principle normal and binormal.
Surfaces were studied by Euler, who investigated curvatures of planar sections and by Gauss, who realized that the product of Euler's two principal curvatures gave a new notion of curvature intrinsic to a surface. Curvature was ultimately extended by Riemann to higher dimensions, and plays today a major role in modern physics, due to the work of Einstein.
If you like this topic, and want to learn more, make sure you don't miss Wildberger's exciting new course on Differential Geometry! See the Playlist DiffGeom, at this channel.
************************
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger
My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated
***********************
Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: https://www.youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************
And here are the Wild Egg Maths Playlists:
Triangle Centres: https://www.youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
м
- published: 07 May 2012
- views: 167147
47:44
Lecture 2B: Introduction to Manifolds (Discrete Differential Geometry)
- Order: Reorder
- Duration: 47:44
- Uploaded Date: 06 Feb 2021
- views: 46689
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
https://wn.com/Lecture_2B_Introduction_To_Manifolds_(Discrete_Differential_Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
- published: 06 Feb 2021
- views: 46689
1:07:46
Lecture 1: Overview (Discrete Differential Geometry)
- Order: Reorder
- Duration: 1:07:46
- Uploaded Date: 04 Feb 2021
- views: 53198
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
https://wn.com/Lecture_1_Overview_(Discrete_Differential_Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
- published: 04 Feb 2021
- views: 53198
8:34
Curvature: Intuition and Derivation | Differential Geometry
- Order: Reorder
- Duration: 8:34
- Uploaded Date: 25 Mar 2020
- views: 20192
In my 5th video on #DifferentialGeometry, I define the #Curvature for both a unit speed curve reparametrized with respect to arc length and a regular curve para...
In my 5th video on #DifferentialGeometry, I define the #Curvature for both a unit speed curve reparametrized with respect to arc length and a regular curve parameterized by t.
I describe the intuition behind the curvature as the extent to which a curve deviates from a straight line (zero curvature). I then derive the expression for curvature for both a unit speed curve and a regular curve, using the #UnitNormal.
Questions/feedback? Let me know in the comments!
Pre-reqs: the previous videos in my playlist https://www.youtube.com/playlist?list=PLdgVBOaXkb9DJjk8V0-RkXTnD4ZXUOFsc
Lecture Notes: https://drive.google.com/open?id=1_40zI8E2r81zOmb_nkhWbN7q2gSqUc7x
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special Thanks to my Patrons:
Cesar Garza
Daigo Saito
Alvin Barnabas
Damjan
Yenyo Pal
Lisa Bouchard
Patapom
Gabriel Sommer
Eugene Bulkin
Yiu Chong
René Gastelumendi
https://wn.com/Curvature_Intuition_And_Derivation_|_Differential_Geometry
In my 5th video on #DifferentialGeometry, I define the #Curvature for both a unit speed curve reparametrized with respect to arc length and a regular curve parameterized by t.
I describe the intuition behind the curvature as the extent to which a curve deviates from a straight line (zero curvature). I then derive the expression for curvature for both a unit speed curve and a regular curve, using the #UnitNormal.
Questions/feedback? Let me know in the comments!
Pre-reqs: the previous videos in my playlist https://www.youtube.com/playlist?list=PLdgVBOaXkb9DJjk8V0-RkXTnD4ZXUOFsc
Lecture Notes: https://drive.google.com/open?id=1_40zI8E2r81zOmb_nkhWbN7q2gSqUc7x
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special Thanks to my Patrons:
Cesar Garza
Daigo Saito
Alvin Barnabas
Damjan
Yenyo Pal
Lisa Bouchard
Patapom
Gabriel Sommer
Eugene Bulkin
Yiu Chong
René Gastelumendi
- published: 25 Mar 2020
- views: 20192
45:20
Lecture 5: Differential Forms (Discrete Differential Geometry)
- Order: Reorder
- Duration: 45:20
- Uploaded Date: 12 Feb 2021
- views: 28232
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
https://wn.com/Lecture_5_Differential_Forms_(Discrete_Differential_Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
- published: 12 Feb 2021
- views: 28232
10:25
Introduction to Differential Geometry: Curves
In this video, I introduce Differential Geometry by talking about curves. Curves and surfa...
published: 10 Jun 2018
Introduction to Differential Geometry: Curves
Introduction to Differential Geometry: Curves
- Report rights infringement
- published: 10 Jun 2018
- views: 160446
In this video, I introduce Differential Geometry by talking about curves. Curves and surfaces are the two foundational structures for differential geometry, which is why I'm introducing this series by defining curves.
After defining level curves, parametrized curves, and tangent vectors, I solve a short example where I convert a level curve to a parametrized curve and then find its tangent vector.
Questions/requests? Let me know in the comments!
Pre-requisites: A background in Multivariable Calculus (Calculus 3) is helpful, but even if you know the material until Calculus 2, you probably still won't be lost.
Lecture Notes: https://drive.google.com/open?id=1CirfXRYfjS8eKB7TVwEWkAT-8nTzpFEQ
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- Yuan Gao
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- Marcin Maciejewski
- Sabre
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- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
1:29:40
Differential Geometry - Claudio Arezzo - Lecture 01
published: 17 Nov 2016
Differential Geometry - Claudio Arezzo - Lecture 01
Differential Geometry - Claudio Arezzo - Lecture 01
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- published: 17 Nov 2016
- views: 159616
51:32
Differential Geometry | Math History | NJ Wildberger
Differential geometry arises from applying calculus and analytic geometry to curves and su...
published: 07 May 2012
Differential Geometry | Math History | NJ Wildberger
Differential Geometry | Math History | NJ Wildberger
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- published: 07 May 2012
- views: 167147
Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the work of C. Huygens on involutes and evolutes, and the related notions of curvature and osculating circle. We discuss involutes of the catenary (yielding the tractrix), cycloid and parabola. The evolute of the parabola is a semi-cubical parabola. For space curves we describe the tangent line, osculating plane, principle normal and binormal.
Surfaces were studied by Euler, who investigated curvatures of planar sections and by Gauss, who realized that the product of Euler's two principal curvatures gave a new notion of curvature intrinsic to a surface. Curvature was ultimately extended by Riemann to higher dimensions, and plays today a major role in modern physics, due to the work of Einstein.
If you like this topic, and want to learn more, make sure you don't miss Wildberger's exciting new course on Differential Geometry! See the Playlist DiffGeom, at this channel.
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Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger
My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated
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Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: https://www.youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
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And here are the Wild Egg Maths Playlists:
Triangle Centres: https://www.youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
м
47:44
Lecture 2B: Introduction to Manifolds (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
Fo...
published: 06 Feb 2021
Lecture 2B: Introduction to Manifolds (Discrete Differential Geometry)
Lecture 2B: Introduction to Manifolds (Discrete Differential Geometry)
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- published: 06 Feb 2021
- views: 46689
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
1:07:46
Lecture 1: Overview (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
Fo...
published: 04 Feb 2021
Lecture 1: Overview (Discrete Differential Geometry)
Lecture 1: Overview (Discrete Differential Geometry)
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- published: 04 Feb 2021
- views: 53198
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
8:34
Curvature: Intuition and Derivation | Differential Geometry
In my 5th video on #DifferentialGeometry, I define the #Curvature for both a unit speed cu...
published: 25 Mar 2020
Curvature: Intuition and Derivation | Differential Geometry
Curvature: Intuition and Derivation | Differential Geometry
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- published: 25 Mar 2020
- views: 20192
In my 5th video on #DifferentialGeometry, I define the #Curvature for both a unit speed curve reparametrized with respect to arc length and a regular curve parameterized by t.
I describe the intuition behind the curvature as the extent to which a curve deviates from a straight line (zero curvature). I then derive the expression for curvature for both a unit speed curve and a regular curve, using the #UnitNormal.
Questions/feedback? Let me know in the comments!
Pre-reqs: the previous videos in my playlist https://www.youtube.com/playlist?list=PLdgVBOaXkb9DJjk8V0-RkXTnD4ZXUOFsc
Lecture Notes: https://drive.google.com/open?id=1_40zI8E2r81zOmb_nkhWbN7q2gSqUc7x
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special Thanks to my Patrons:
Cesar Garza
Daigo Saito
Alvin Barnabas
Damjan
Yenyo Pal
Lisa Bouchard
Patapom
Gabriel Sommer
Eugene Bulkin
Yiu Chong
René Gastelumendi
45:20
Lecture 5: Differential Forms (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
Fo...
published: 12 Feb 2021
Lecture 5: Differential Forms (Discrete Differential Geometry)
Lecture 5: Differential Forms (Discrete Differential Geometry)
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- published: 12 Feb 2021
- views: 28232
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to defining integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.
For instance, the expression f(x) dx from one-variable calculus is called a 1-form, and can be integrated over an interval [a, b] in the domain of f:
and similarly the expression f(x, y, z) dx ∧ dy + g(x, y, z) dx ∧ dz + h(x, y, z) dy ∧ dz is a 2-form that has a surface integral over an oriented surface S:
Likewise, a 3-form f(x, y, z) dx ∧ dy ∧ dz represents a volume element that can be integrated over a region of space.
The algebra of differential forms is organized in a way that naturally reflects the orientation of the domain of integration. There is an operation d on differential forms known as the exterior derivative that, when acting on a k-form, produces a (k + 1)-form. This operation extends the differential of a function, and the divergence and the curl of a vector field in an appropriate sense that makes the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem special cases of the same general result, known in this context also as the general Stokes' theorem. In a deeper way, this theorem relates the topology of the domain of integration to the structure of the differential forms themselves; the precise connection is known as de Rham's theorem.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Differential_form
Introduction to Differential Geometry: Curves...
Differential Geometry - Claudio Arezzo - Lecture 0...
Differential Geometry | Math History | NJ Wildberg...
Lecture 2B: Introduction to Manifolds (Discrete Di...
Lecture 1: Overview (Discrete Differential Geometr...
Curvature: Intuition and Derivation | Differential...