excellent examples that brilliantly show the proper application of the push-forward. looking forward to seeing the application to GR which professionally covers the subject.
Another fine lesson. Thank you. Two questions about notation Please correct me if I'm wrong. 1) At 20:45 or so, it could be clearer if you included the third component with respect to the third, d/d(sigma_3), basis vector in the first line, even though it has zero coefficient. This would emphasize that N is a 3 dimensional space and its tangent space is 3 dimensional, too. At first I was confused that there were only two components. 2) At 8:25 or so, you have a notation Df(p) 'dot' v_p. I think it means that the matrix Df, evaluated at point p is multiplying the vector v_p. This is matrix multiplication, not a dot product of vectors, though so it might be a little confusing to notate it this way as a dot product. Or is this a notation for matrix multiplication, too? BTW: I think I have seen Df(p) called the 'differential' of f at p in some texts (Spivak's 'Calculus on Manifolds'?), rather than the 'Jacobian Matrix' of the function f.. I believe this is a shift in the usual meaning of 'differential'. In calculus the 'differential' of a function usually means a linearized increment of a function (corresponding to some implicit increments of its inputs). In contrast, here Df would be the function taking those input increments that computes the corresponding increment of the output function, and these can have explicit values. I understand this wasn't the terminology you chose in this lesson, but thanks if you could help correct me or clarify. Thanks again.
Thank you for the kind words and for your detailed questions! Let me address each of your points: 1. Components in the Tangent Space Basis (20:45) You're absolutely correct that explicitly including the third component (even if it has a coefficient of zero) would emphasize that both 𝑁 and its tangent space 𝑇_𝑝𝑁 are 3-dimensional. This is a great point about clarity, especially when communicating to an audience that might be working through the nuances of dimensions in tangent spaces. I'll make a note to adjust this in future presentations to avoid any ambiguity. Including all components explicitly helps reinforce the dimensionality of the tangent space, even if some coefficients are zero. 2. Notation 𝐷𝑓(𝑝)⋅𝑣_𝑝 (8:25) Great observation here! The notation 𝐷𝑓(𝑝)⋅𝑣_𝑝 is used to denote the matrix multiplication of the Jacobian matrix 𝐷𝑓(𝑝) with the tangent vector 𝑣_𝑝. You're right that this could be confusing since a "dot" often suggests the dot product between two vectors in other contexts. In this case, the dot doesn't represent the usual Euclidean inner product but rather the application of the linear map 𝐷𝑓(𝑝) (viewed as a matrix) to the vector 𝑣_𝑝 . Your suggestion to clarify the notation is well-taken; perhaps using something like 𝐷𝑓(𝑝)(𝑣_𝑝) could help avoid confusion in future discussions. 3. The Term "Differential" vs. "Jacobian Matrix" You're absolutely correct about the terminology. In many texts, including Spivak's Calculus on Manifolds, 𝐷𝑓(𝑝) is referred to as the "differential" of 𝑓 at 𝑝, emphasizing its interpretation as a linear map. This aligns with the geometric idea that 𝐷𝑓(𝑝) is the best linear approximation of 𝑓 at the point 𝑝. The distinction you mention between the "differential" as a linear map and the Jacobian matrix as its representation in coordinates is a subtle but important one. In the context of multivariable calculus, the Jacobian matrix is often introduced as the matrix of partial derivatives, while the term "differential" highlights the more abstract, coordinate-independent perspective. I appreciate you bringing up this point, and I’ll make an effort to clarify these nuances in future lessons. Final Thoughts Thank you again for such thoughtful questions and comments! They not only highlight important nuances but also help improve how these ideas are communicated. If you have any further questions, feel free to ask!
Thank you so much for these valuable efforts Mr Davie. I have a question: what is the difference between the Pushforward operation and coordinate transformation?
Thank you so much for your kind words and thoughtful question! Let’s break this down and clarify the difference between the pushforward operation and a coordinate transformation: 1. Pushforward Operation: The pushforward is a mathematical operation associated with a map between two manifolds, say 𝑓:𝑀→𝑁. If you have a tangent vector 𝑋_𝑝 at a point 𝑝∈𝑀, the pushforward 𝑓_∗ maps 𝑋_𝑝 to a tangent vector 𝑓_∗(𝑋_𝑝) at 𝑓(𝑝)∈𝑁. This is a way to "transport" the action of a vector field from the domain manifold 𝑀 to the codomain manifold 𝑁, based on the behavior of the function 𝑓. To understand it intuitively, if 𝑋_𝑝 represents a direction and rate of change of functions on 𝑀, then 𝑓_∗(𝑋_𝑝) represents how those changes translate to 𝑁 via the map 𝑓. The pushforward depends not on a coordinate system but on the geometry of 𝑓 itself. 2. Coordinate Transformation: A coordinate transformation, on the other hand, is a change of the local coordinate system used to describe the same manifold. For example, you might switch from Cartesian coordinates (𝑥,𝑦) to polar coordinates (𝑟,𝜃) on a 2D plane. Under a coordinate transformation, the components of vectors, tensors, or forms change according to specific transformation rules (e.g., chain rule for vectors). However, the geometric object itself remains the same-only its representation in terms of coordinates changes. 3. Key Difference: - Pushforward: Transfers tangent vectors (or vector fields) from one manifold to another via a map 𝑓:𝑀→𝑁. It involves two different manifolds. - Coordinate Transformation: Changes the representation of vectors or tensors on the same manifold by switching between different coordinate systems. To summarize: - Pushforward deals with mapping vectors between manifolds. - Coordinate transformation deals with describing vectors differently on the same manifold. I hope this clears things up! If you’d like a specific example or further clarification, feel free to ask. Thank you for watching and engaging with these lessons!
excellent examples that brilliantly show the proper application of the push-forward. looking forward to seeing the application to GR which professionally covers the subject.
Another fine lesson. Thank you.
Two questions about notation Please correct me if I'm wrong.
1) At 20:45 or so, it could be clearer if you included the third component with respect to the third, d/d(sigma_3), basis vector in the first line, even though it has zero coefficient. This would emphasize that N is a 3 dimensional space and its tangent space is 3 dimensional, too. At first I was confused that there were only two components.
2) At 8:25 or so, you have a notation Df(p) 'dot' v_p. I think it means that the matrix Df, evaluated at point p is multiplying the vector v_p. This is matrix multiplication, not a dot product of vectors, though so it might be a little confusing to notate it this way as a dot product. Or is this a notation for matrix multiplication, too?
BTW: I think I have seen Df(p) called the 'differential' of f at p in some texts (Spivak's 'Calculus on Manifolds'?), rather than the 'Jacobian Matrix' of the function f.. I believe this is a shift in the usual meaning of 'differential'. In calculus the 'differential' of a function usually means a linearized increment of a function (corresponding to some implicit increments of its inputs). In contrast, here Df would be the function taking those input increments that computes the corresponding increment of the output function, and these can have explicit values.
I understand this wasn't the terminology you chose in this lesson, but thanks if you could help correct me or clarify.
Thanks again.
Thank you for the kind words and for your detailed questions! Let me address each of your points:
1. Components in the Tangent Space Basis (20:45)
You're absolutely correct that explicitly including the third component (even if it has a coefficient of zero) would emphasize that both 𝑁 and its tangent space
𝑇_𝑝𝑁 are 3-dimensional. This is a great point about clarity, especially when communicating to an audience that might be working through the nuances of dimensions in tangent spaces. I'll make a note to adjust this in future presentations to avoid any ambiguity. Including all components explicitly helps reinforce the dimensionality of the tangent space, even if some coefficients are zero.
2. Notation 𝐷𝑓(𝑝)⋅𝑣_𝑝 (8:25)
Great observation here! The notation 𝐷𝑓(𝑝)⋅𝑣_𝑝 is used to denote the matrix multiplication of the Jacobian matrix 𝐷𝑓(𝑝) with the tangent vector 𝑣_𝑝. You're right that this could be confusing since a "dot" often suggests the dot product between two vectors in other contexts. In this case, the dot doesn't represent the usual Euclidean inner product but rather the application of the linear map 𝐷𝑓(𝑝) (viewed as a matrix) to the vector 𝑣_𝑝 . Your suggestion to clarify the notation is well-taken; perhaps using something like 𝐷𝑓(𝑝)(𝑣_𝑝) could help avoid confusion in future discussions.
3. The Term "Differential" vs. "Jacobian Matrix"
You're absolutely correct about the terminology. In many texts, including Spivak's Calculus on Manifolds,
𝐷𝑓(𝑝) is referred to as the "differential" of 𝑓 at 𝑝, emphasizing its interpretation as a linear map. This aligns with the geometric idea that 𝐷𝑓(𝑝) is the best linear approximation of 𝑓 at the point 𝑝.
The distinction you mention between the "differential" as a linear map and the Jacobian matrix as its representation in coordinates is a subtle but important one. In the context of multivariable calculus, the Jacobian matrix is often introduced as the matrix of partial derivatives, while the term "differential" highlights the more abstract, coordinate-independent perspective. I appreciate you bringing up this point, and I’ll make an effort to clarify these nuances in future lessons.
Final Thoughts
Thank you again for such thoughtful questions and comments! They not only highlight important nuances but also help improve how these ideas are communicated. If you have any further questions, feel free to ask!
Thank you so much for these valuable efforts Mr Davie.
I have a question: what is the difference between the Pushforward operation and coordinate transformation?
Thank you so much for your kind words and thoughtful question! Let’s break this down and clarify the difference between the pushforward operation and a coordinate transformation:
1. Pushforward Operation: The pushforward is a mathematical operation associated with a map between two manifolds, say 𝑓:𝑀→𝑁. If you have a tangent vector 𝑋_𝑝 at a point 𝑝∈𝑀, the pushforward
𝑓_∗ maps 𝑋_𝑝 to a tangent vector 𝑓_∗(𝑋_𝑝) at 𝑓(𝑝)∈𝑁. This is a way to "transport" the action of a vector field from the domain manifold 𝑀 to the codomain manifold
𝑁, based on the behavior of the function 𝑓.
To understand it intuitively, if 𝑋_𝑝 represents a direction and rate of change of functions on 𝑀, then 𝑓_∗(𝑋_𝑝) represents how those changes translate to 𝑁 via the map 𝑓. The pushforward depends not on a coordinate system but on the geometry of 𝑓 itself.
2. Coordinate Transformation: A coordinate transformation, on the other hand, is a change of the local coordinate system used to describe the same manifold. For example, you might switch from Cartesian coordinates (𝑥,𝑦) to polar coordinates (𝑟,𝜃) on a 2D plane. Under a coordinate transformation, the components of vectors, tensors, or forms change according to specific transformation rules (e.g., chain rule for vectors). However, the geometric object itself remains the same-only its representation in terms of coordinates changes.
3. Key Difference:
- Pushforward: Transfers tangent vectors (or vector fields) from one manifold to another via a map 𝑓:𝑀→𝑁. It involves two different manifolds.
- Coordinate Transformation: Changes the representation of vectors or tensors on the same manifold by switching between different coordinate systems.
To summarize:
- Pushforward deals with mapping vectors between manifolds.
- Coordinate transformation deals with describing vectors differently on the same manifold.
I hope this clears things up! If you’d like a specific example or further clarification, feel free to ask. Thank you for watching and engaging with these lessons!
4:58 a=p?
Yes to that. Sorry for the confusion.