Thanks. Good lecture. Question: In physics undergrad we had a lot of awkward formulas for expressing the divergence, gradient and curl in other coordinate systems (like spherical coordinates) besides the standard Euclidean ones. The derivations of these formulas were always awkward and either used hand wavy physical arguments or dense pages of vector calculus manipulations. Can you use the pullback formula to do this job more elegantly? Can you interpret the divergence, gradient, curl as forms on one space and pull back to express it as a form in another space? The spaces would really the same in this case, just Euclidean 3 space in Euclidean versus spherical or cylindrical coordinates.. Thanks if you can comment.
Thank you for your question! You're absolutely right that the traditional derivations of divergence, gradient, and curl in spherical or cylindrical coordinates can feel awkward or overly complicated. Using differential forms and the pullback operation offers a more elegant and geometrically motivated framework to approach these concepts. Let me explain how this can work. 1. Understanding Divergence, Gradient, and Curl as Forms In the language of differential forms: The gradient corresponds to a 1-form because it measures the directional rate of change of a scalar function. The curl corresponds to a 2-form, which captures rotational flow (it can also be represented as a pseudovector in 3D). The divergence corresponds to a 3-form, representing the "volume density" of a flux. Using the exterior derivative 𝑑, we can express: Gradient: For a scalar function 𝑓, ∇𝑓 corresponds to the 1-form 𝑑𝑓. Curl: For a vector field 𝐹, ∇×𝐹 corresponds to ⋆𝑑𝜔, where 𝜔 is the 1-form dual to 𝐹. Divergence: For a vector field 𝐹, ∇⋅𝐹 corresponds to ⋆𝑑⋆𝜔 , where 𝜔 is again the 1-form dual to 𝐹. All of the above will be covered in future videos. 2. Pullbacks and Coordinate Transformations The pullback operation 𝜙∗ can be used to relate forms between coordinate systems. Suppose you have a coordinate transformation 𝜙 : 𝑈 ⊂ 𝑅^3 → 𝑉⊂𝑅^3 , such as the mapping from Euclidean coordinates (𝑥 , 𝑦, 𝑧) to spherical coordinates (𝑟, 𝜃, 𝜙). The pullback 𝜙∗ maps forms in one coordinate system to forms in the other. Key idea: The pullback automatically incorporates the coordinate transformation and adjusts the components and basis of the form appropriately. This allows you to derive expressions for divergence, gradient, and curl in the new coordinates without hand-waving or tedious vector calculus manipulations. 3. Example: Gradient in Spherical Coordinates Let’s compute the gradient of a scalar function 𝑓 in spherical coordinates using pullbacks. Start with 𝑑𝑓 in Euclidean Coordinates: In Cartesian coordinates, 𝑑𝑓 = ∂𝑓/∂𝑥 𝑑𝑥 + ∂𝑓/∂𝑦 𝑑𝑦 + ∂𝑓/∂𝑧 𝑑𝑧 . Express the Coordinate Transformation: Spherical coordinates are related to Cartesian coordinates by: 𝑥 = 𝑟sin𝜃cos𝜙 , 𝑦 = 𝑟sin𝜃sin𝜙 , 𝑧 = 𝑟cos𝜃 . The inverse transformation is: 𝑟 = sqrt[𝑥^2 + 𝑦^2 + 𝑧^2] , 𝜃 = arccos(𝑧/𝑟) , 𝜙 = arctan(𝑦/𝑥). Pullback the 1-Form 𝑑𝑓 to Spherical Coordinates: Using the pullback, the 1-form 𝑑𝑓 transforms into spherical coordinates as: 𝑑𝑓 = ∂𝑓/∂𝑟 𝑑𝑟 + ∂𝑓/∂𝜃 𝑑𝜃 + ∂𝑓/∂𝜙 𝑑𝜙 . Interpretation: This result gives the gradient in spherical coordinates directly in terms of the coordinate basis 𝑑𝑟 , 𝑑𝜃 , 𝑑𝜙 bypassing the need for tedious recalculations. 4. Extending to Curl and Divergence For the curl and divergence, you can follow similar steps: Use the Hodge star ⋆ operator to express them in terms of exterior derivatives and wedge products. Pull back the forms to the new coordinates. The pullback naturally incorporates the metric (via the volume form vol = sqrt[∣𝑔∣] 𝑑𝑟∧𝑑𝜃∧𝑑𝜙 and ensures that the resulting expressions are correct for the transformed coordinates. 5. Advantages of Using Pullbacks Geometric Insight: Pullbacks highlight the intrinsic nature of these operations, showing how they depend only on the geometry of the space, not the specific choice of coordinates. Streamlined Derivations: Instead of manipulating individual components of vectors and tensors, pullbacks handle the transformations for you, leaving you with clean and elegant results. Summary: Yes, the pullback formula can be used to express the gradient, divergence, and curl in alternative coordinate systems more elegantly. By treating these operations as differential forms and transforming them using pullbacks, you avoid the messy vector calculus manipulations and gain a deeper geometric understanding of the process. Thank you for your insightful question-feel free to ask if you'd like more details or a specific example worked out in full! I will be covering all of this in future videos.
@@TensorCalculusRobertDavie Thank you Dr. Davie. You have an opportunity here to really do a good deed. Much of the teaching of this material in mainstream sources that undergrad and beginning grad students have to use just isn't very good. The main reason is that no text is perfect and the deficiencies in teaching and in textbooks need to be corrected. Unfortunately, that doesn't happen. Instead, bad approaches often become accepted tradition from year to year, generation to generation. The people who 'get it' gloat, the people who don't feel to embarrassed to speak up. Someone needs to take a fresh look at how this material is taught, streamline it, clarify it conceptually, and patch up the weaknesses and holes. You seem like a great guy to contribute to that important work. Speaking personally, I had to give up math and physics after undergrad and never fully came to peace with why it had to be that way. Half a lifetime later, and seeing the material explained clearly on TH-cam, it makes me realize that perhaps it wasn't entirely my fault. Makes me hope that maybe my son will have a better experience!
@@johnwarren8032 Hello John, and thank you for your comment. I found it quite moving as it brought back memories of my younger self and the difficulties I faced. You are absolutely right about textbooks and their limitations. Authors are restricted to providing a brief overview of the subject and often have limited time to write. This constraint means that including the numerous diagrams you see on my channel for each concept is not feasible in textbooks. Such additions would result in very large and potentially prohibitively expensive volumes for students. For these reasons, and others, TH-camrs like myself can complement textbooks by making the concepts they cover more accessible through detailed and carefully thought-out explanations. As you rightly pointed out, how concepts are explained is crucial and can make the difference between success and failure in learning. In fact, a single well-explained concept can significantly ease access to other ideas in the same field. Clear explanations matter. I think we can all agree that anything carefully and accurately explained can be understood by almost everyone. We see examples of this in the world around us, and mathematics and science are no exceptions, as your post correctly highlights. It all comes down to presentation. It was this passion for clear explanation, along with a love of learning, that drove me to become a teacher. I am not a Dr. The university exam system does what it is designed for-funneling the most capable students into higher research degrees-and, by doing so, makes a significant contribution to society. However, learning neither begins nor ends with university; it is an enabler of something inherent in each of us: a desire to learn. We don't need higher degrees to affirm our interest in higher-level mathematics, physics, or any other field of inquiry. It’s simply a matter of perseverance-putting one foot in front of the other. Everything else is secondary.
A few side questions (my terminology may be way off it’s been 10 years since I’ve touched anything serious). Does this work if the main surface M. Was let’s say smoothly bumpy and you choose a p that is in a valley of the surface M. Does this still work. With N and M . (Or do N and M have to be locally the same “shape” in the area that the operation is happening. ) Does it not matter since we are only concerned with a point p on the surface? Would the tangent plane intersecting the main object outside of point p even matter? Ty for the video.
if you take a step back, you can realise that none of these operations are actually dependent on the geometric structure of the manifold. a manifold just needs to be locally (diffeo-) homeomorphic to euclidean space, so only the (local) topology matters. the geometry of M and N is actually induced or defined by differential forms and tensors, instead of the other way around.
Thank you for your thoughtful questions! I’ll break down the key concepts you’re asking about and how they relate to the operations being performed on the surface 𝑀. 1. Does the Smoothness of 𝑀 Matter? If 𝑀 is smoothly bumpy and you choose a point 𝑝 in a valley of the surface, the overall approach still works. The key idea is that we are only concerned with local properties at the point 𝑝. Regardless of how the surface behaves elsewhere, as long as 𝑀 is smooth at 𝑝, we can define a well-behaved tangent plane at that point. The surface can have bumps, valleys, or peaks, but as long as 𝑀 is smooth, meaning it has continuous derivatives at 𝑝, we can still perform the same operations. The tangent plane at 𝑝 will be a good approximation of 𝑀 near 𝑝, even if the surface curves elsewhere. 2. Do 𝑀 and 𝑁 Need to Be Locally the Same Shape? No, 𝑀 and 𝑁 (if 𝑁 is another surface or manifold you're working with) don’t need to have the same local shape for operations like taking tangent planes or projecting vectors to work. The operations you're concerned with typically happen in the tangent space at 𝑝, which is inherently linear and independent of the larger geometry of the surfaces themselves. As long as both 𝑀 and 𝑁 are smooth at the point of interest, we can work with their tangent planes at 𝑝. The overall shapes of 𝑀 and 𝑁 are less important for local operations at 𝑝. 3. Does the Tangent Plane Intersecting the Object Elsewhere Matter? No, the intersection of the tangent plane with the surface outside of point 𝑝 doesn’t affect the operations at 𝑝. The tangent plane is only an approximation that "touches" the surface at 𝑝 and provides local linear behavior, but it’s not meant to describe the global structure of 𝑀. The tangent plane is a local object, and what happens far from 𝑝 is irrelevant for operations that only depend on the behavior near 𝑝. Conclusion: In summary, the operations you’re referring to focus on local properties near the point 𝑝, so the global shape of the surface doesn’t interfere. As long as 𝑀 is smooth at 𝑝, and both 𝑀 and 𝑁 are smooth in the neighborhood of 𝑝, these operations will work as intended. I hope this clears up your questions, and feel free to ask if you’d like further clarification!
Thank you! So glad to see some deep diving into differential geometry. Looking forward to more!
Than you for that!
Thanks. Good lecture. Question: In physics undergrad we had a lot of awkward formulas for expressing the divergence, gradient and curl in other coordinate systems (like spherical coordinates) besides the standard Euclidean ones. The derivations of these formulas were always awkward and either used hand wavy physical arguments or dense pages of vector calculus manipulations.
Can you use the pullback formula to do this job more elegantly? Can you interpret the divergence, gradient, curl as forms on one space and pull back to express it as a form in another space? The spaces would really the same in this case, just Euclidean 3 space in Euclidean versus spherical or cylindrical coordinates..
Thanks if you can comment.
Thank you for your question! You're absolutely right that the traditional derivations of divergence, gradient, and curl in spherical or cylindrical coordinates can feel awkward or overly complicated. Using differential forms and the pullback operation offers a more elegant and geometrically motivated framework to approach these concepts. Let me explain how this can work.
1. Understanding Divergence, Gradient, and Curl as Forms
In the language of differential forms:
The gradient corresponds to a 1-form because it measures the directional rate of change of a scalar function.
The curl corresponds to a 2-form, which captures rotational flow (it can also be represented as a pseudovector in 3D).
The divergence corresponds to a 3-form, representing the "volume density" of a flux.
Using the exterior derivative 𝑑, we can express:
Gradient: For a scalar function 𝑓, ∇𝑓 corresponds to the 1-form 𝑑𝑓.
Curl: For a vector field 𝐹, ∇×𝐹 corresponds to ⋆𝑑𝜔, where 𝜔 is the 1-form dual to 𝐹.
Divergence: For a vector field 𝐹, ∇⋅𝐹 corresponds to ⋆𝑑⋆𝜔 , where 𝜔 is again the 1-form dual to 𝐹.
All of the above will be covered in future videos.
2. Pullbacks and Coordinate Transformations
The pullback operation 𝜙∗ can be used to relate forms between coordinate systems. Suppose you have a coordinate transformation 𝜙 : 𝑈 ⊂ 𝑅^3 → 𝑉⊂𝑅^3 , such as the mapping from Euclidean coordinates
(𝑥 , 𝑦, 𝑧) to spherical coordinates (𝑟, 𝜃, 𝜙). The pullback
𝜙∗ maps forms in one coordinate system to forms in the other.
Key idea:
The pullback automatically incorporates the coordinate transformation and adjusts the components and basis of the form appropriately.
This allows you to derive expressions for divergence, gradient, and curl in the new coordinates without hand-waving or tedious vector calculus manipulations.
3. Example: Gradient in Spherical Coordinates
Let’s compute the gradient of a scalar function 𝑓 in spherical coordinates using pullbacks.
Start with 𝑑𝑓 in Euclidean Coordinates: In Cartesian coordinates, 𝑑𝑓 = ∂𝑓/∂𝑥 𝑑𝑥 + ∂𝑓/∂𝑦 𝑑𝑦 + ∂𝑓/∂𝑧 𝑑𝑧 .
Express the Coordinate Transformation: Spherical coordinates are related to Cartesian coordinates by:
𝑥 = 𝑟sin𝜃cos𝜙 , 𝑦 = 𝑟sin𝜃sin𝜙 , 𝑧 = 𝑟cos𝜃 .
The inverse transformation is:
𝑟 = sqrt[𝑥^2 + 𝑦^2 + 𝑧^2] , 𝜃 = arccos(𝑧/𝑟) ,
𝜙 = arctan(𝑦/𝑥).
Pullback the 1-Form 𝑑𝑓 to Spherical Coordinates: Using the pullback, the 1-form 𝑑𝑓 transforms into spherical coordinates as:
𝑑𝑓 = ∂𝑓/∂𝑟 𝑑𝑟 + ∂𝑓/∂𝜃 𝑑𝜃 + ∂𝑓/∂𝜙 𝑑𝜙 .
Interpretation: This result gives the gradient in spherical coordinates directly in terms of the coordinate basis
𝑑𝑟 , 𝑑𝜃 , 𝑑𝜙 bypassing the need for tedious recalculations.
4. Extending to Curl and Divergence
For the curl and divergence, you can follow similar steps:
Use the Hodge star ⋆ operator to express them in terms of exterior derivatives and wedge products.
Pull back the forms to the new coordinates.
The pullback naturally incorporates the metric (via the volume form vol = sqrt[∣𝑔∣] 𝑑𝑟∧𝑑𝜃∧𝑑𝜙
and ensures that the resulting expressions are correct for the transformed coordinates.
5. Advantages of Using Pullbacks
Geometric Insight: Pullbacks highlight the intrinsic nature of these operations, showing how they depend only on the geometry of the space, not the specific choice of coordinates.
Streamlined Derivations: Instead of manipulating individual components of vectors and tensors, pullbacks handle the transformations for you, leaving you with clean and elegant results.
Summary: Yes, the pullback formula can be used to express the gradient, divergence, and curl in alternative coordinate systems more elegantly. By treating these operations as differential forms and transforming them using pullbacks, you avoid the messy vector calculus manipulations and gain a deeper geometric understanding of the process.
Thank you for your insightful question-feel free to ask if you'd like more details or a specific example worked out in full! I will be covering all of this in future videos.
@@TensorCalculusRobertDavie Thank you Dr. Davie. You have an opportunity here to really do a good deed. Much of the teaching of this material in mainstream sources that undergrad and beginning grad students have to use just isn't very good. The main reason is that no text is perfect and the deficiencies in teaching and in textbooks need to be corrected. Unfortunately, that doesn't happen. Instead, bad approaches often become accepted tradition from year to year, generation to generation. The people who 'get it' gloat, the people who don't feel to embarrassed to speak up. Someone needs to take a fresh look at how this material is taught, streamline it, clarify it conceptually, and patch up the weaknesses and holes. You seem like a great guy to contribute to that important work.
Speaking personally, I had to give up math and physics after undergrad and never fully came to peace with why it had to be that way. Half a lifetime later, and seeing the material explained clearly on TH-cam, it makes me realize that perhaps it wasn't entirely my fault. Makes me hope that maybe my son will have a better experience!
@@johnwarren8032 Hello John, and thank you for your comment. I found it quite moving as it brought back memories of my younger self and the difficulties I faced.
You are absolutely right about textbooks and their limitations. Authors are restricted to providing a brief overview of the subject and often have limited time to write. This constraint means that including the numerous diagrams you see on my channel for each concept is not feasible in textbooks. Such additions would result in very large and potentially prohibitively expensive volumes for students.
For these reasons, and others, TH-camrs like myself can complement textbooks by making the concepts they cover more accessible through detailed and carefully thought-out explanations. As you rightly pointed out, how concepts are explained is crucial and can make the difference between success and failure in learning. In fact, a single well-explained concept can significantly ease access to other ideas in the same field. Clear explanations matter.
I think we can all agree that anything carefully and accurately explained can be understood by almost everyone. We see examples of this in the world around us, and mathematics and science are no exceptions, as your post correctly highlights. It all comes down to presentation.
It was this passion for clear explanation, along with a love of learning, that drove me to become a teacher. I am not a Dr.
The university exam system does what it is designed for-funneling the most capable students into higher research degrees-and, by doing so, makes a significant contribution to society. However, learning neither begins nor ends with university; it is an enabler of something inherent in each of us: a desire to learn.
We don't need higher degrees to affirm our interest in higher-level mathematics, physics, or any other field of inquiry. It’s simply a matter of perseverance-putting one foot in front of the other. Everything else is secondary.
A few side questions (my terminology may be way off it’s been 10 years since I’ve touched anything serious). Does this work if the main surface M. Was let’s say smoothly bumpy and you choose a p that is in a valley of the surface M. Does this still work. With N and M . (Or do N and M have to be locally the same “shape” in the area that the operation is happening. ) Does it not matter since we are only concerned with a point p on the surface? Would the tangent plane intersecting the main object outside of point p even matter? Ty for the video.
if you take a step back, you can realise that none of these operations are actually dependent on the geometric structure of the manifold. a manifold just needs to be locally (diffeo-) homeomorphic to euclidean space, so only the (local) topology matters. the geometry of M and N is actually induced or defined by differential forms and tensors, instead of the other way around.
@@shum8104 Thank you for that!
Thank you for your thoughtful questions! I’ll break down the key concepts you’re asking about and how they relate to the operations being performed on the surface 𝑀.
1. Does the Smoothness of 𝑀 Matter?
If 𝑀 is smoothly bumpy and you choose a point
𝑝 in a valley of the surface, the overall approach still works. The key idea is that we are only concerned with local properties at the point 𝑝. Regardless of how the surface behaves elsewhere, as long as 𝑀 is smooth at 𝑝, we can define a well-behaved tangent plane at that point.
The surface can have bumps, valleys, or peaks, but as long as 𝑀 is smooth, meaning it has continuous derivatives at 𝑝, we can still perform the same operations.
The tangent plane at 𝑝 will be a good approximation of 𝑀 near 𝑝, even if the surface curves elsewhere.
2. Do 𝑀 and 𝑁 Need to Be Locally the Same Shape?
No, 𝑀 and 𝑁 (if 𝑁 is another surface or manifold you're working with) don’t need to have the same local shape for operations like taking tangent planes or projecting vectors to work. The operations you're concerned with typically happen in the tangent space at 𝑝, which is inherently linear and independent of the larger geometry of the surfaces themselves.
As long as both 𝑀 and 𝑁 are smooth at the point of interest, we can work with their tangent planes at 𝑝. The overall shapes of 𝑀 and 𝑁 are less important for local operations at 𝑝.
3. Does the Tangent Plane Intersecting the Object Elsewhere Matter?
No, the intersection of the tangent plane with the surface outside of point 𝑝 doesn’t affect the operations at 𝑝. The tangent plane is only an approximation that "touches" the surface at 𝑝 and provides local linear behavior, but it’s not meant to describe the global structure of 𝑀.
The tangent plane is a local object, and what happens far from 𝑝 is irrelevant for operations that only depend on the behavior near 𝑝.
Conclusion:
In summary, the operations you’re referring to focus on local properties near the point 𝑝, so the global shape of the surface doesn’t interfere. As long as 𝑀 is smooth at 𝑝, and both 𝑀 and 𝑁 are smooth in the neighborhood of 𝑝, these operations will work as intended.
I hope this clears up your questions, and feel free to ask if you’d like further clarification!