big picture of vector calculus - math 212 · big picture of vector calculus math 212 brian d....
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![Page 1: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/1.jpg)
Big Picture of Vector CalculusMath 212
Brian D. Fitzpatrick
Duke University
April 21, 2020
MATH
![Page 2: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/2.jpg)
Overview
Summary of ConstructionsShapes in R3
Exact SequencesVector Calculus Theorems and Exact Sequences
![Page 3: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/3.jpg)
Summary of ConstructionsShapes in R3
ObservationIn R3, we have four types of shapes.
points curvessurfaces
solids
The boundary of a shape is a lower-dimensional shape.
(points)∂←− (curves)
∂←− (surfaces)∂←− (solids)
The boundary of a shape is “closed” so this sequence is exact.
![Page 4: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/4.jpg)
Summary of ConstructionsShapes in R3
ObservationIn R3, we have four types of shapes.
points
curvessurfaces
solids
The boundary of a shape is a lower-dimensional shape.
(points)∂←− (curves)
∂←− (surfaces)∂←− (solids)
The boundary of a shape is “closed” so this sequence is exact.
![Page 5: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/5.jpg)
Summary of ConstructionsShapes in R3
ObservationIn R3, we have four types of shapes.
points curves
surfacessolids
The boundary of a shape is a lower-dimensional shape.
(points)∂←− (curves)
∂←− (surfaces)∂←− (solids)
The boundary of a shape is “closed” so this sequence is exact.
![Page 6: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/6.jpg)
Summary of ConstructionsShapes in R3
ObservationIn R3, we have four types of shapes.
points curvessurfaces
solids
The boundary of a shape is a lower-dimensional shape.
(points)∂←− (curves)
∂←− (surfaces)∂←− (solids)
The boundary of a shape is “closed” so this sequence is exact.
![Page 7: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/7.jpg)
Summary of ConstructionsShapes in R3
ObservationIn R3, we have four types of shapes.
points curvessurfaces
solids
The boundary of a shape is a lower-dimensional shape.
(points)∂←− (curves)
∂←− (surfaces)∂←− (solids)
The boundary of a shape is “closed” so this sequence is exact.
![Page 8: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/8.jpg)
Summary of ConstructionsShapes in R3
ObservationIn R3, we have four types of shapes.
points curvessurfaces
solids
The boundary of a shape is a lower-dimensional shape.
(points)∂←− (curves)
∂←− (surfaces)∂←− (solids)
The boundary of a shape is “closed” so this sequence is exact.
![Page 9: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/9.jpg)
Summary of ConstructionsShapes in R3
ObservationIn R3, we have four types of shapes.
points curvessurfaces
solids
The boundary of a shape is a lower-dimensional shape.
(points)∂←− (curves)
∂←− (surfaces)
∂←− (solids)
The boundary of a shape is “closed” so this sequence is exact.
![Page 10: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/10.jpg)
Summary of ConstructionsShapes in R3
ObservationIn R3, we have four types of shapes.
points curvessurfaces
solids
The boundary of a shape is a lower-dimensional shape.
(points)∂←− (curves)
∂←− (surfaces)∂←− (solids)
The boundary of a shape is “closed” so this sequence is exact.
![Page 11: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/11.jpg)
Summary of ConstructionsShapes in R3
ObservationIn R3, we have four types of shapes.
points curvessurfaces
solids
The boundary of a shape is a lower-dimensional shape.
(points)
∂←− (curves)∂←− (surfaces)
∂←− (solids)
The boundary of a shape is “closed” so this sequence is exact.
![Page 12: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/12.jpg)
Summary of ConstructionsShapes in R3
ObservationIn R3, we have four types of shapes.
points curvessurfaces
solids
The boundary of a shape is a lower-dimensional shape.
(points)∂←− (curves)
∂←− (surfaces)∂←− (solids)
The boundary of a shape is “closed” so this sequence is exact.
![Page 13: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/13.jpg)
Summary of ConstructionsShapes in R3
ObservationIn R3, we have four types of shapes.
points curvessurfaces
solids
The boundary of a shape is a lower-dimensional shape.
(points)∂←− (curves)
∂←− (surfaces)∂←− (solids)
The boundary of a shape is “closed” so this sequence is exact.
![Page 14: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/14.jpg)
Summary of ConstructionsExact Sequences
ObservationIn R3, we have the operator exact sequence.
C (R3)grad−−→ X(R3)
curl−−→ X(R3)div−−→ C (R3)
Somehow, these two sequences “match up.”
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂ ∂
grad curl div
There are three “sections” of these sequences.
![Page 15: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/15.jpg)
Summary of ConstructionsExact Sequences
ObservationIn R3, we have the operator exact sequence.
C (R3)grad−−→ X(R3)
curl−−→ X(R3)div−−→ C (R3)
Somehow, these two sequences “match up.”
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂ ∂
grad curl div
There are three “sections” of these sequences.
![Page 16: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/16.jpg)
Summary of ConstructionsExact Sequences
ObservationIn R3, we have the operator exact sequence.
C (R3)grad−−→ X(R3)
curl−−→ X(R3)div−−→ C (R3)
Somehow, these two sequences “match up.”
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂ ∂
grad curl div
There are three “sections” of these sequences.
![Page 17: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/17.jpg)
Summary of ConstructionsExact Sequences
ObservationIn R3, we have the operator exact sequence.
C (R3)grad−−→ X(R3)
curl−−→ X(R3)div−−→ C (R3)
Somehow, these two sequences “match up.”
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂ ∂
grad curl div
There are three “sections” of these sequences.
![Page 18: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/18.jpg)
Summary of ConstructionsExact Sequences
ObservationIn R3, we have the operator exact sequence.
C (R3)grad−−→ X(R3)
curl−−→ X(R3)div−−→ C (R3)
Somehow, these two sequences “match up.”
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂ ∂
grad curl div
There are three “sections” of these sequences.
![Page 19: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/19.jpg)
Summary of ConstructionsExact Sequences
ObservationIn R3, we have the operator exact sequence.
C (R3)grad−−→ X(R3)
curl−−→ X(R3)div−−→ C (R3)
Somehow, these two sequences “match up.”
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂∂ ∂ ∂
grad curl div
There are three “sections” of these sequences.
![Page 20: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/20.jpg)
Summary of ConstructionsExact Sequences
ObservationIn R3, we have the operator exact sequence.
C (R3)grad−−→ X(R3)
curl−−→ X(R3)div−−→ C (R3)
Somehow, these two sequences “match up.”
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂ ∂
grad curl div
There are three “sections” of these sequences.
![Page 21: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/21.jpg)
Summary of ConstructionsExact Sequences
ObservationIn R3, we have the operator exact sequence.
C (R3)grad−−→ X(R3)
curl−−→ X(R3)div−−→ C (R3)
Somehow, these two sequences “match up.”
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂∂ ∂
grad curl div
There are three “sections” of these sequences.
![Page 22: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/22.jpg)
Summary of ConstructionsExact Sequences
ObservationIn R3, we have the operator exact sequence.
C (R3)grad−−→ X(R3)
curl−−→ X(R3)div−−→ C (R3)
Somehow, these two sequences “match up.”
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂ ∂
grad curl div
There are three “sections” of these sequences.
![Page 23: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/23.jpg)
Summary of ConstructionsVector Calculus Theorems and Exact Sequences
ObservationIncidentally, we also have three vector calculus theorems.
Fundamental Theorem of Line IntegralsˆCgrad(f ) · ds = f (Q)− f (P)
Stokes’ Theorem¨Scurl(F ) · dS =
˛∂S
F · ds
Divergence Theorem˚Ddiv(F ) dV =
‹∂D
F · dS
Each theorem “pairs” with a section of our exact sequences!
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂ ∂
grad curl div
![Page 24: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/24.jpg)
Summary of ConstructionsVector Calculus Theorems and Exact Sequences
ObservationIncidentally, we also have three vector calculus theorems.
Fundamental Theorem of Line IntegralsˆCgrad(f ) · ds = f (Q)− f (P)
Stokes’ Theorem¨Scurl(F ) · dS =
˛∂S
F · ds
Divergence Theorem˚Ddiv(F ) dV =
‹∂D
F · dS
Each theorem “pairs” with a section of our exact sequences!
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂ ∂
grad curl div
![Page 25: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/25.jpg)
Summary of ConstructionsVector Calculus Theorems and Exact Sequences
ObservationIncidentally, we also have three vector calculus theorems.
Fundamental Theorem of Line IntegralsˆCgrad(f ) · ds = f (Q)− f (P)
Stokes’ Theorem¨Scurl(F ) · dS =
˛∂S
F · ds
Divergence Theorem˚Ddiv(F ) dV =
‹∂D
F · dS
Each theorem “pairs” with a section of our exact sequences!
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂ ∂
grad curl div
![Page 26: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/26.jpg)
Summary of ConstructionsVector Calculus Theorems and Exact Sequences
ObservationIncidentally, we also have three vector calculus theorems.
Fundamental Theorem of Line IntegralsˆCgrad(f ) · ds = f (Q)− f (P)
Stokes’ Theorem¨Scurl(F ) · dS =
˛∂S
F · ds
Divergence Theorem˚Ddiv(F ) dV =
‹∂D
F · dS
Each theorem “pairs” with a section of our exact sequences!
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂ ∂
grad curl div
![Page 27: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/27.jpg)
Summary of ConstructionsVector Calculus Theorems and Exact Sequences
ObservationIncidentally, we also have three vector calculus theorems.
Fundamental Theorem of Line IntegralsˆCgrad(f ) · ds = f (Q)− f (P)
Stokes’ Theorem¨Scurl(F ) · dS =
˛∂S
F · ds
Divergence Theorem˚Ddiv(F ) dV =
‹∂D
F · dS
Each theorem “pairs” with a section of our exact sequences!
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂ ∂
grad curl div
![Page 28: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/28.jpg)
Summary of ConstructionsVector Calculus Theorems and Exact Sequences
ObservationIncidentally, we also have three vector calculus theorems.
Fundamental Theorem of Line IntegralsˆCgrad(f ) · ds = f (Q)− f (P)
Stokes’ Theorem¨Scurl(F ) · dS =
˛∂S
F · ds
Divergence Theorem˚Ddiv(F ) dV =
‹∂D
F · dS
Each theorem “pairs” with a section of our exact sequences!
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂∂ ∂ ∂
grad curl div
![Page 29: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/29.jpg)
Summary of ConstructionsVector Calculus Theorems and Exact Sequences
ObservationIncidentally, we also have three vector calculus theorems.
Fundamental Theorem of Line IntegralsˆCgrad(f ) · ds = f (Q)− f (P)
Stokes’ Theorem¨Scurl(F ) · dS =
˛∂S
F · ds
Divergence Theorem˚Ddiv(F ) dV =
‹∂D
F · dS
Each theorem “pairs” with a section of our exact sequences!
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂ ∂
grad curl div
![Page 30: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/30.jpg)
Summary of ConstructionsVector Calculus Theorems and Exact Sequences
ObservationIncidentally, we also have three vector calculus theorems.
Fundamental Theorem of Line IntegralsˆCgrad(f ) · ds = f (Q)− f (P)
Stokes’ Theorem¨Scurl(F ) · dS =
˛∂S
F · ds
Divergence Theorem˚Ddiv(F ) dV =
‹∂D
F · dS
Each theorem “pairs” with a section of our exact sequences!
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂∂ ∂
grad curl div
![Page 31: Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D. Fitzpatrick Duke University April 21, 2020 MATH. Overview Summary of Constructions Shapes](https://reader030.vdocument.in/reader030/viewer/2022041109/5f0dc01e7e708231d43be6bb/html5/thumbnails/31.jpg)
Summary of ConstructionsVector Calculus Theorems and Exact Sequences
ObservationIncidentally, we also have three vector calculus theorems.
Fundamental Theorem of Line IntegralsˆCgrad(f ) · ds = f (Q)− f (P)
Stokes’ Theorem¨Scurl(F ) · dS =
˛∂S
F · ds
Divergence Theorem˚Ddiv(F ) dV =
‹∂D
F · dS
Each theorem “pairs” with a section of our exact sequences!
(points) (curves) (surfaces) (solids)
C (R3) X(R3) X(R3) C (R3)
∂ ∂ ∂
grad curl div