Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D....
Transcript of Big Picture of Vector Calculus - Math 212 · Big Picture of Vector Calculus Math 212 Brian D....
Big Picture of Vector CalculusMath 212
Brian D. Fitzpatrick
Duke University
April 21, 2020
MATH
Overview
Summary of ConstructionsShapes in R3
Exact SequencesVector Calculus Theorems and Exact Sequences
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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
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
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
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
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
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
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
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