ma 105 : calculus division 3, lecture 30 · 2019-10-20 · division 3, lecture 30 prof. sudhir r....

MA 105 : Calculus

Division 3, Lecture 30

Prof. Sudhir R. GhorpadeIIT Bombay

Prof. Sudhir R. Ghorpade, IIT Bombay MA 105 Calculus: Division 3, Lecture 30

Recap of the previous lecture

Basic facts about triple integrals

Fubini’s theorem on cuboids

Sets of (three-dimensional) content zero. Examples

Criterion for the integrability of a function of 3 variables

Elemenentary regions. Cavalieri Principle

Volume of a bouded subset of the 3-space

Domain additivity for triple integrals

Change of Variables in a Triple Integral. Example

Cylindrical and Spherical coordinates. Examples.

Vector algebra

Notions of cross product, scalar triple product and vectortriple product


Scalar Triple Product and Vector triple Product

Let xxx , yyy , zzz ∈ R3. Then xxx · (yyy × zzz) ∈ R and xxx × (yyy × zzz) ∈ R3

are called the scalar triple product and the vector tripleproduct of xxx , yyy , zzz respectively. It is easy to see that ifxxx := (x1, x2, x3), yyy := (y1, y2, y3), zzz := (z1, z2, z3), then

xxx · (yyy × zzz) =

∣∣∣∣∣∣x1 x2 x3y1 y2 y3z1 z2 z3

∣∣∣∣∣∣ .Geometrically, xxx · (yyy × zzz) can be interpreted as the (signed)volume of the parallelopiped defined by the vectors xxx ,yyy ,zzz .

One can prove the Lagrange formula

xxx × (yyy × zzz) = (xxx · zzz)yyy − (xxx · yyy)zzz

by considering each component of the LHS and the RHS.Prof. Sudhir R. Ghorpade, IIT Bombay MA 105 Calculus: Division 3, Lecture 30

Scalar Fields and Vector Fields

A scalar field is an assignment of a scalar to each point in aregion in the space. For example, the temperature at a pointon the earth is a scalar field (defined on a subset of R3).

A vector field is an assignment of a vector to each point in aregion in the space. For example, the velocity field of amoving fluid is a vector field that associates a velocity vectorto each point in the fluid.

Definition

Let m ∈ N, and let D be a subset of Rm.

A scalar field is a function from D to R.

A vector field is a function from D to Rm.If m = 2, then it is called a vector field in the plane, andif m = 3, then it is called a vector field in the space.


Depiction of Vector Fields

FFF (x , y) := (2x , 2y) FFF (x , y) := (−x ,−y)/√x2 + y 2



FFF (x , y) := (y , x) FFF (x , y) := (−y , x)



FFF (x , y) := (sin y , cos x)


Smooth Scalar and Vector Fields

Suppose D is an open subset of Rm, that is, every point in Dis an interior point of D.

A scalar field f : D → R is called smooth if∂f

∂xjexists and is

continuous on D for j = 1, . . . ,m. The set of all smoothscalar fields on D is denoted by C 1(D). Similarly, the set of allscalar fields on D having continuous partial derivatives of thefirst and second order is denoted by C 2(D).

Let FFF : D → Rm be a vector field on D, and let

FFF (xxx) := (F1(xxx), . . . ,Fm(xxx)) for xxx ∈ D,

where Fi : D → R is the ith component scalar field on Dfor i = 1, . . . ,m. The vector field FFF is called smooth on D if

each Fi is smooth on D, that is,∂Fi

∂xjexists and is continuous

on D for i , j = 1, . . . ,m.Prof. Sudhir R. Ghorpade, IIT Bombay MA 105 Calculus: Division 3, Lecture 30

Gradient, Divergence and Curl

Let D be an open subset of R3.If f is a smooth scalar field defined onD, then the vector field

grad f := ∇f =

(∂f

∂x,∂f

∂y,∂f

∂z

)defined on D is called the gradient field of f .

If FFF := (P ,Q,R) is a smooth vector field defined on D, thenthe divergence field of FFF is the scalar field on D defined by

divFFF := ∇ · FFF =∂P

∂x+∂Q

∂y+∂R

∂z,

and the curl field of FFF is the vector field on D defined by

curlFFF := ∇× FFF =

(∂R

∂y− ∂Q

∂z,∂P

∂z− ∂R

∂x,∂Q

∂x− ∂P

∂y

).


As in the case of the cross product xxx × yyy , we may write

∇× FFF = ∇× (P ,Q,R) =

∣∣∣∣∣∣∣∣iii jjj kkk∂

∂x

∂

∂y

∂

∂zP Q R

∣∣∣∣∣∣∣∣ .For any m ∈ N, we can define the gradient field grad f of ascalar field f defined on an open subset of Rm, and also thedivergence field divFFF of a vector field FFF defined on an opensubset of Rm in a similar manner. But the curl field curlFFF isdefined only for a vector field FFF on an open subset of R3.

If D ⊂ R is open, and f is a smooth scalar field on D, then for(x , y , z)∈R×D×R, we can let FFF (x , y , z) :=(0, f (x), 0) anddefine curl f :=curlFFF =(0, 0, f ′). Also, if D ⊂ R2 is open andΦΦΦ := (P ,Q) is a smooth vector field on D, then for(x , y , z) ∈ D×R, we can let FFF (x , y , z) := (P(x , y),Q(x , y), 0)and define curlΦΦΦ := curlFFF = (0, 0,Qx − Py ).


GCD Sequence

Let D be an open subset of R3. Suppose the first and thesecond order partial derivatives of f ,P ,Q,R : D → R existand are continuous on D. By the Mixed Partials Theorem,(i) curl (grad f ) = ∇× (∇f ) = 000:(∂2f

∂y∂z− ∂2f

∂z∂y,∂2f

∂z∂x− ∂2f

∂x∂z,∂2f

∂x∂y− ∂2f

∂y∂x

)= (0, 0, 0),

(ii)and if FFF := (P ,Q,R), then div (curlFFF ) = ∇ · (∇× FFF ) = 0:

∂

∂x

(∂R

∂y− ∂Q

∂z

)+

∂

∂y

(∂P

∂z− ∂R

∂x

)+

∂

∂z

(∂Q

∂x− ∂P

∂y

)= 0.

Gradient, Curl and Divergence are three operators{scalarfields

}grad−→

{vectorfields

}curl−→

{vectorfields

}div−→

{scalarfields

}which satisfy curl (grad f ) = 000 and div (curlFFF ) = 0, so thatthe successive composites are zero.


The above phenomenon raises the following basic questions:

(i) If GGG is a smooth vector field such that curlGGG = 000, thenmust GGG be a gradient field, that is, is there a scalar field fsuch that GGG = grad f ? (Then f is called a potential field.)

(ii) If HHH is a smooth vector field such that divHHH = 0, thenmust HHH be a curl field, that is, is there a vector field FFF suchthat HHH = curlFFF?

These questions remind us of the following situation. Considerthe operator der : C 1([a, b])→ C ([a, b]) given by der(f ) = f ′.If g is a continuous function on [a, b], then is there acontinuously differentiable function f on [a, b] such thatg = f ′? The Fundamental theorem of Calculus (Part I)answers this question in the affirmative: If we letf (x) :=

∫ x

ag(t)dt for x ∈ [a, b], then f ′ = g on [a, b].

In view of this, each of the two questions raised above call fora suitable theory of integration, to which we now turn.


Laplacian

Let D be an open subset of R3, and let f be a smooth scalarfield defined on D. Suppose the second order partialderivatives fxx , fyy , fzz exist on D. The Laplacian field of f isthe scalar field defined on D by

div (grad f ) := ∇ · (∇f ) = ∇2f =∂2f

∂x2+∂2f

∂y 2+∂2f

∂z2.

The Laplacian field is the composition of the operators gradand div: {

scalarfields

}grad−→

{vectorfields

}div−→

{scalarfields

}.

The Laplacian plays a very important role in the theory ofpartial differential equations, and its various applications.


Paths

Let α, β ∈ R with α < β, and let m ∈ N. A path or aparametrized curve in Rm is a continuous functionγγγ : [α, β]→ Rm, that is, if γγγ := (γ1, . . . , γm), then thefunction γj : [α, β]→ R is continuous for each j = 1, . . . ,m.

A path γγγ : [α, β]→ Rm is called closed if γγγ(α) = γγγ(β). Apath γγγ : [α, β]→ Rm is called simple if γγγ(t1) 6= γγγ(t2) fort1 < t2 in [α, β] unless t1 = α and t2 = β.

Let t ∈ [α, β] and suppose γ′1(t), . . . , γ′m(t) exist. Then

dγγγ

dt= γγγ′(t) := (γ′1(t), . . . , γ′m(t))

is called the tangent vector to γγγ at t. If γγγ′(t) 6= 000, thent̂tt := γγγ′(t)/‖γγγ′(t)‖ is called the unit tangent vector to γγγ att. (We write t̂tt instead of t̂ttγγγ(t) for brevity.)


Further, a path γγγ in Rm is called smooth if eachγj : [α, β]→ R is continuously differentiable for j = 1, . . . ,m;in case γγγ is a closed curve, that is, if γγγ(α) = γγγ(β), we alsorequire γγγ′(α) = γγγ′(β). A smooth path is also called a C 1-path.

A smooth path γγγ in Rm is called regular if γγγ′(t) 6= 000 for eacht ∈ [α, β], that is, if the unit tangent vector to γγγ exists ateach t ∈ [α, β].

Examples: For t ∈ [−1, 1], let γγγ1(t) := (t, t2) andγγγ2(t) := (t2, t3). Then γγγ1 is regular, but γγγ2 is not.

A path γγγ in Rm is called piecewise smooth if there areα := t0 < t1 < · · · < tn =: β such that γγγ is smooth on each[ti−1, ti ], i = 1, . . . , n, and it is called piecewise regular, if γγγis regular on each [ti−1, ti ], i = 1, . . . , n.

We shall assume hereafter that all paths are piecewise smooth,unless otherwise stated.


Example Let a > 0. Define γγγ(t) := (a cos t, a sin t) fort ∈ [−π, π]. This path is called the standard parametrizedcircle in R2 of radius a. Since γγγ(−π) = (−1, 0) = γγγ(π) andγγγ′(t) := (−a sin t, a cos t) for t ∈ [−π, π], we see that thepath γγγ is closed, simple, regular, and its unit tangent vector att ∈ [−π, π] is t̂tt := (− sin t, cos t).

If we let γ̃γγ(t) := (a cos t,−a sin t) for t ∈ [−π, π], then it iseasy to see that the path γ̃γγ is also closed, simple, regular, andits unit tangent vector at t ∈ [−π, π] is (− sin t,− cos t).

Note that γ̃γγ([−π, π]) = γγγ([−π, π]), that is, the functions γ̃γγand γγγ have the same range, although they are clearly differentpaths: one goes around the circle anticlockwise, but the othergoes around the circle clockwise.

Further, the path given by γ̃γγ(t) := (a cos 2t, a sin 2t) fort ∈ [−π, π] is closed and regular, its unit tangent vector att ∈ [−π, π] is (− sin 2t, cos 2t), and has the same range as γγγ.


Path-connected and Convex Subsets

A subset D of Rm is called path-connected if for alluuu,vvv ∈ D, there is a path γγγ : [α, β]→ Rm such thatγγγ(α) = uuu, γγγ(β) = vvv and γγγ(t) ∈ D for all t ∈ (α, β).

A subset D of Rm is called convex if the line segment joiningany two points in D lies in D. A convex set is path-connectedsince for uuu,vvv ∈ D, the straight-line path γγγ : [0, 1]→ Rm

defined by γγγ(t) := uuu + t(vvv − uuu) for t ∈ [0, 1], lies in D.

Examples:(i) The subset {(x , y) ∈ R2 : x2 + y 2 ≤ 1} of R2 ispath-connected; in fact it is convex.

(ii) The subset {(x , y) ∈ R2 : 1/2 ≤ x2 + y 2 ≤ 1} of R2 ispath-connected, but it is not convex.

(iii) The subset {(x , y) ∈ R2 : x2 + y 2 ≤ 1} ∪ {(2, 0)} of R2 isnot path-connected.


Arc length

Let γγγ := (γ1, . . . , γm) : [α, β]→ Rm be a piecewise smoothparametrized curve. We define the arc length of γγγ by

`(γγγ) :=

∫ β

α

‖γγγ′(t)‖dt =

∫ β

α

√γ′1(t)2 + · · ·+ γ′m(t)2dt.

This definition agrees with our earlier definition if m = 2, 3.

ExampleLet γγγ denote the standard parametrized circle in R2 of radius

a. Then `(γγγ) =∫ π

−π

√a2 sin2 t + a2 cos2 t dt = 2πa.

Next, let γ̃γγ(t) := (a cos 2t, a sin 2t) for t ∈ [−π, π]. Then

`(γ̃γγ) =∫ π

−π

√4a2 sin2 2t + 4a2 cos2 2t dt = 4πa.

Note that γ̃γγ([−π, π]) = γγγ([−π, π]), but `(γ̃γγ) = 2`(γγγ).


Line Integral of a Scalar Field

Let m ∈ N. Let γγγ : [α, β]→ Rm be a (piecewise smooth)path, and let C := γγγ([α, β]). Let f : C → R be a boundedscalar field. We define the line integral of f along γγγ by∫

γγγ

f ds :=

∫ β

α

f (γγγ(t))‖γγγ′(t)‖dt,

provided the Riemann integral on the right side exists. Inparticular, if f is continuous, then

∫γγγf ds is well-defined.

Letting f := 1 on C , we see that∫γγγds = `(γγγ).

Example: Let f (x , y) := x2 + y 2 for (x , y) ∈ R2. If γγγ is thestandard parametrized circle of radius a, then∫

γγγ

f ds =

∫ π

−π(a2 cos2 t + a2 sin2 t)a dt = 2πa3.


ma 105 : calculus division 3, lecture 30 · 2019-10-20 · division 3, lecture 30 prof. sudhir r....

Documents