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