Fundamental Theorems

Chapter 5

5.1 Revisiting the 1D Fundamental Theorem of Calculus

Key Ideas.

Alternate take on the derivative of a function f : R R. • ! – The function f determines a one-dimensional vector ﬁeld V~ = f = f@ = fi. h i x – Let Br =[p r, p + r] be the interval of “radius” r centered at point p . ⇤ ⇤ ⇤ – The boundary of Br consists of two points p + r and p r.The“outwardunitnormal” ⇤ ⇤ to Br is given by @x at x = p + r Nˆ = ⇤ ( @x at x = p r. ⇤

– The outﬂux of V~ across the boundary of Br is

V~ Nˆ = f(p + r) f(p r). · ⇤ ⇤ boundaryX points f(p + r) f(p r) – The outﬂux per unit size is ⇤ ⇤ 2r f(p + r) f(p r) – Linear approximation gives us ⇤ ⇤ = f 0(p )+error, where error 0. 2r ⇤ ! Thus the derivative of f at p is the limit of the “e↵ect per unit size” of the function. • ⇤ The Fundamental Theorem of Calculus can be obtained by summing over small intervals. . . • Our plan is now to ﬁnd notions of derivatives, and corresponding fundamental theorems, for • various quantities. . .

73 74 CHAPTER 5. FUNDAMENTAL THEOREMS

5.2 Fundamental Theorem for Gradients

Key Ideas.

Consider a function f : Rn R and a path P (t)=(x(t),y(t),...) deﬁned for a t b. Let • !   A = P (a) and B = P (b). We have f P :[a, b] R. By the 1D Fundamental Theorem of Calculus we have ! b (f P )0(t) dt = f(A) f(B). Za From the Chain Rule we have • @f dx @x(P (t)) dt @f dy (f P )0(t)=[Df(P (t))]P 0(t)=0 @y (P (t))1 0 dt 1 · . . B . C B . C B . C B . C @ A @ A Recall that the gradient of a function f: • @f @f grad(f)= , ,... . @x @y ⌧ Geometric idea of gradient: The gradient of a function measures the total change of a function. • If Aˆ is any unit vector, then grad(f) Aˆ tells us the extent to which f changes in that direction. (This is the same as the directional derivative.)· The gradient vector points in the direction of largest increase, with magnitude equal to the rate of change in that direction. Di↵erentiation rules: • – Scaling rule: grad(cf)=c grad(f) if c is a constant – Addition rule: grad(f + g) = grad(f) + grad(g) – Product rule: grad fg = grad(f)g + f grad(g)

Fundamental Theorem of Calculus for gradients:Supposef : Rn R is smooth1 and • that C is a parametrized path from point A to point B. Then !

grad(f) Tdsˆ = f(B) f(A). · ZC Exercises. 1. Compute the gradient for the following functions: (a) f(x, y)=x2 + y2 (b) f(x, y)=x2 y2 (c) f(x, y)=ln(x2 + y2)

1In this course smooth means that the object is continuous and that derivatrives of the object are also continuous. 5.2. FUNDAMENTAL THEOREM FOR GRADIENTS 75

(d) f(x, y, z)= x2 + y2 + z2 2. Can you ﬁnd a functionp f so that. . . (a) . . . grad(f)= 2, 3 ? h i (b) . . . grad(f)= x, y ? h i (c) . . . grad(f)= y, x ? h i (d) . . . grad(f)= y, x ? h i 3. Let C be the path traveling counter-clockwise along the unit circle from (1, 0) to ( 1, 0). Let L be the path traveling along the x axis from (1, 0) to ( 1, 0). (a) Find parametrizations for C and L. (b) Compute x, y Tdsˆ by direct methods. C h i· (c) Compute R x, y Tdsˆ by direct methods. Lh i· (d) Find a function f such that grad(f)= x, y . R h i (e) Compute x, y Tdsˆ using the Fundamental Theorem for Gradients. C h i· (f) Compute R x, y Tdsˆ using the Fundamental Theorem for Gradients. Lh i· (g) Do your computationsR agree? Explain why/why not. 4. Let C be the path traveling counter-clockwise along the unit circle from (1, 0) to ( 1, 0). Let C be the path traveling clockwise along the unit circle from (1, 0) to ( 1, 0). e(a) Find parametrizations of C and C. (b) Compute y, x Tdsˆ by direct methods. C h i· e (c) Compute R y, x Tdsˆ by direct methods. C h i· (d) Do your computations agree? Explain why/why not. R e 76 CHAPTER 5. FUNDAMENTAL THEOREMS

5.3 Divergence of a vector ﬁeld

Key Ideas.

Geometric idea of divergence: The divergence of vector ﬁeld V~ at the point P measures the • expansion of V~ at P ⇤ ⇤

Deﬁnition of divergence using limits: Let Br be the solid ball of radius r centered at point P • ⇤ and Sr be the boundary of Br. 1 1 div V~ (P )=lim V~ NdAˆ and/or div V~ (P )=lim V~ Ndsˆ ⇤ r 0 B · ⇤ r 0 B · ! | r| ZZSr ! | r| ZSr

Computation in Cartesian coordinates: Suppose V~ = V ,V ,... . Then • h x y i @V @V div V~ = x + y + .... @x @y

Di↵erentiation rules: • – Scaling rule: div(cV~ )=c div V~ if c is a constant – Addition rule: div(V~ + W~ )=divV~ +divW~ – Product rule: div(fV~ ) = grad(f) V~ + f div V~ if f is a function · Exercises. 1. For each vector field do the following: make a sketch of the vector field by hand, describe in words what the vector field does, compute the divergence of the vector field, comment on how your computation fits with your earlier description.

(a) V~ = x, y h i (b) V~ = y, x h i (c) V~ = 3, 2 h i (d) V~ = x , y h x2+y2 x2+y2 i (e) V~ = x y, x + y h i (f) V~ = x + y, x y h i (g) V~ = y2,y2 h i 2. Suppose f(x, y)=x2 y2 and V~ = x + y, x +2y . h i (a) Compute fV~ and then compute div(fV~ ). (b) Use the product rule to compute div(fV~ ). (c) Prove the product rule formula for divergence. 5.3. DIVERGENCE OF A VECTOR FIELD 77

3. The Laplacian of a function f is defined by div(grad f). Compute the Laplacian of the following functions. (a) f(x, y)=2x 3y (b) f(x, y)=x2 + y2 (c) f(x, y)=x2 y2 (d) f(x, y) = cos(x)sin(y) 1 2 2 (e) f(x, y)= 2 ln(x + y ) 1 (f) f(x, y, z)= x2 + y2 + z2 What property ofp the function does the Laplacian seem to measure? 4. Consider the vector field given in spherical coordinates by 1 x y z V~ = @ = , , r2 r (x2 + y2 + z2)3/2 (x2 + y2 + z2)3/2 (x2 + y2 + z2)3/2 D E and concentric spheres Sr of radius r centered at the origin. (a) Compute the outward flux V~ NdAˆ using the definition of the flux integral; Sr · (b) Find the flux rate RR V~ NdAˆ lim Sr · r 0 Volume(B ) ! RR r where Br denotes the ball enclosed by Sr. (c) Compute the divergence of the vector field V~ . (d) What is the outward flux rate of V~ at the origin? What about anywhere other than the origin? 5. (Optional Challenge Problem) In this problem you consider the possibility of defining divergence of a vector field V~ (x, y) using a square, rather than a ball. Fill in the boxes in the argument below.

(a) For simplicity we work in two dimensions. Write V~ (x, y)= Vx(x, y),Vy(x, y) .Fixa point (x ,y ). For small x and y we have h i ⇤ ⇤ @V @V V~ (x, y) V~ (x ,y )+(x x ) x (x ,y ), y (x ,y ) ⇡ ⇤ ⇤ ⇤ @x ⇤ ⇤ @x ⇤ ⇤ ⌧

+(y y ) , ⇤ * +

2 (b) Let Br denote the interior of the square in R having corners at (x + r, y + r), (x ⇤ ⇤ ⇤ r, y + r), (x r, y r), (x + r, y r). Let Sr be the boundary of Br, oriented counter-clockwise.⇤ ⇤ ⇤ ⇤ ⇤ 78 CHAPTER 5. FUNDAMENTAL THEOREMS

Draw a picture showing Br and Sr.

The area of Br is

(c) The outward unit normal Nˆ along Sr is given by 1, 0 when x = x + r, h i ⇤ 8 when y = y + r, > ⇤ > ˆ > N = > > when , <>

> > when , > > > (d) The integral V~ Ndsˆ can:> be approximated by Sr ·

r R r V~ (x + r, y + t) 1, 0 dt + 0, 1 dt t= r ⇤ ⇤ ·h i t=r ·h i Z Z

r + dt + . Zt=r

(e) Using the approximation from the ﬁrst part of this problem we we ﬁnd that r @V @V V~ Ndsˆ V~ (x ,y )+r x (x ,y )+t x (x ,y ) dt S · ⇡ t= r ⇤ ⇤ @x ⇤ ⇤ @y ⇤ ⇤ Z r Z ✓ ◆ r + V~ (x ,y )+t + r dt 0 ⇤ ⇤ 1 Zt=r B C @ A r V~ (x ,y )+r t dt 0 ⇤ ⇤ 1 Zt=r B C @ A r dt t= r Z

(f) The previous integral simpliﬁes to

@V V~ Ndsˆ x (x ,y )+ . · ⇡ 0 @x ⇤ ⇤ 1 ZSr B C @ A 5.3. DIVERGENCE OF A VECTOR FIELD 79

Thus V~ Ndsˆ @V lim Sr · = x (x ,y )+ r 0 @x ⇤ ⇤ ! R

and we are done! 80 CHAPTER 5. FUNDAMENTAL THEOREMS

5.4 Divergence Theorems

Key Ideas.

Two-dimensional Divergence Theorem. Let D be a bounded domain in R2, let C be the • boundary of D, and let Nˆ be the outward-pointing unit normal along C. Then for any smooth vector ﬁeld V~ deﬁned in D we have

div VdA~ = V~ Nds.ˆ · ZZD ZC The two-dimensional divergence theorem is equivalent to Green’s Theorem. (However, Green’s theorem is usually written in a di↵erent way – see the next section.)

Three-dimensional Divergence Theorem. Let D be a bounded domain in R3, let S be the • boundary of D, and let Nˆ be the outward-pointing unit normal along S. Then for any smooth vector ﬁeld V~ deﬁned in D we have

div VdV~ = V~ Nˆ dA. · ZZZD ZS The three-dimensional divergence theorem is sometimes called Gauss’ Theorem.

Justification: divide D in to many small domains and use the definition of divergence. • Note: The vector field V~ and its divergence need to be defined in all of D. •

Important application: If div V~ =0in a region D whose boundary has two parts: S1 and S2 • then V~ NdAˆ + V~ NdAˆ =0. · · ZZS1 ZZS2 (A similar formula holds if D is a region in R2.) Vector ﬁelds whose divergence is zero are called called divergence-free or incompressible.

Exercises.

1. Consider the circles C0, C1, C2, C3, and the outward pointing vector fields Nˆ as in Figure 1. Suppose V~ (x, y) is a vector field defined (and smooth) throughout the xy-plane. Furthermore, suppose that V~ has the following outward fluxes:

V~ Ndsˆ =1, V~ Ndsˆ =3, V~ Ndsˆ = 2, V~ Ndsˆ =5. · · · · ZC0 ZC1 ZC2 ZC3 ~ Use Green’s Theorem to evaluate D div(V ) dA where D denotes

(a) The area inside C0 but outsideRR C2;

(b) The area inside C0 but outside both C1 and C3. 5.4. DIVERGENCE THEOREMS 81

Nˆ

C1 C3 C2

Nˆ

Figure 5.1: The ﬁgure for homework problem 1.

~ ˆ 2. Evaluate the outward flux integrals C V Ndsfor the following choices of the vector fields ~ · V and the curves C. You are expectedR to use the divergence theorem. (a) V~ (x, y)= 3y, x ,whileC is the counter-clockwise unit circle centered at the origin. h i (b) V~ (x, y)= x2 + xy + y2,x2 xy + y2 ,whileC is the boundary of the rectangle with vertices ath (0, 0), (1, 0), (1, 2), (0, 2). i (c) V~ (x, y)= x, y ,whileC is the boundary of the triangle with vertices at ( 1, 0), (0, 1) and (1, 0).h i 3. Use Gauss’ Theorem to evaluate the following flux integrals.

(a) The outward flux of V~ (x, y, z)= x3 +y3,y3 +z3,z3 +x3 across the unit sphere centered at the origin; h i (b) The inward flux of V~ (x, y, z)= xy, yz, xz across the bowl of the elliptic paraboloid z = x2 + y2 h i i. up to and including the lid at z = 4; ii. up to but not including the lid at z = 4; (c) The inward flux of the vector field V~ (x, y, z)= x2 + x, y2 + y, z2 z across the cone h i z = x2 + y2 i. pup to and including the lid at z = 2; ii. up to but not including the lid at z = 2; (d) The outward flux of V~ (x, y, z)= x, y, z across the cube 1 x, y, z 1. h i   82 CHAPTER 5. FUNDAMENTAL THEOREMS

5.5 Scalar curl; 2D Curl Theorem; Green’s Theorem

Key Ideas. Geometric idea of scalar curl: The curl of a 2D vector ﬁeld V~ at a point P measures the • counter-clockwise rotational work done by V~ at that point. ⇤

Definition of scalar curl using limits: Let Br be the solid disk of radius r centered at P and • ⇤ let Sr be the circular boundary of Br and let Tˆ be the counter-clockwise unit tangent along Sr. The scalar curl of vectorfield V~ is defined by 1 curl(V~ )=lim V~ Tds.ˆ r 0 B · ! | r| ZCr Computation in Cartesian coordinates: Suppose V~ = V ,V . Then curl(V~ )= @Vy @Vx . • h x yi @x @y @ V ~ @x x Computation using determinants: curl V =det @ @y Vy! Di↵erentiation rules: • – Scaling rule: curl(cV~ )=c curl V~ – Addition rule: curl(V~ + W~ )=curlV~ +curlW~ @f V ~ @x x ~ – Product rule: curl(fV )=det @f + f curl V . @y Vy! The two-dimensional Curl Theorem: Let D be a bounded domain in R2, let C be the • boundary of D, and let Tˆ be the counter-clockwise oriented tangent vector along C.Forany smooth vector field V~ defined in D we have

curl VdA~ = V~ Tds.ˆ · ZZD ZC Justiﬁcation of the 2D Curl Theorem: Divide D in to many small squares. . . • Suppose that V~ = P, Q . Then curl(V~ ) dA = @Q @P dxdy.SupposealsothatC is • h i @x @y parametrized by the functions x(t),y(t).Sincedx ⇣= x0(t) and⌘ dy = y0(t) we can write Tdsˆ = dx, dy and thus V~ Tdsˆ = Pdx+ Qdy. Thus the 2D Curl Theorem can be written h i · @Q @P dxdy = Pdx+ Qdy. @x @y ZZD ✓ ◆ ZC This formula is known as Green’s Theorem.

Suppose that W~ = Q, P . Then div(W~ )dA = @Q @P dxdy.WithC parametrized by • h i @x @y the functions x(t),y(t) we have dx = x0(t) and dy⇣ = y0(t) as⌘ above. Thus Ndsˆ = dy, dx h i and the 2D divergence theorem for W~ becomes @Q @P dxdy = Pdx+ Qdy. @x @y ZZD ✓ ◆ ZC Green’s Theorem encapsulates both the 2D Curl Theorem and the 2D Divergence Theorem. 5.5. SCALAR CURL; 2D CURL THEOREM; GREEN’S THEOREM 83

Exercises. 1. Draw a picture of each vector ﬁeld. Use the picture to estimate the scalar curl. Then compute the scalar curl and compare. (a) V~ = x, y h i (b) V~ = y, x h i (c) V~ = y, x h i y x (d) V~ = , x2 + y2 x2 + y2 ⌧ ~ ˆ 2. Use Green’s Theorem to evaluate the circulation integrals C V Tdsfor the following choices ~ · of V and C. R (a) V~ (x, y)= 3y, x ,whileC is the counter-clockwise unit circle centered at the origin. h i (b) V~ (x, y)= x2 + xy + y2,x2 xy + y2 ,whileC is the boundary of the rectangle with vertices ath (0, 0), (1, 0), (1, 2), (0, 2) orientedi counter-clockwise. (c) V~ (x, y)= 3x2y + y3, x3 3xy2 ,whileC is the boundary of the unit circle centered at the originh oriented counter-clockwise. i (d) V~ (x, y)= x, y ,whileC is the boundary of the triangle with vertices at ( 1, 0), (0, 1) and (1, 0) orientedh i clock-wise.

3. Consider the conﬁguration on the following diagram. Note that C5 is the bold path with distinct starting and ending points, and that C6 denotes the “left-over”.

C2 C5

Suppose V~ (x, y) is a vector ﬁeld deﬁned (and smooth) throughout the xy-plane. Furthermore, suppose the following:

V~ Tdsˆ = 1, V~ Tdsˆ =1, V~ Tdsˆ = 2, V~ Tdsˆ =2, V~ Tdsˆ = 3, · · · · · ZC1 ZC2 ZC3 ZC4 ZC5 84 CHAPTER 5. FUNDAMENTAL THEOREMS

while curl(V~ ) vanishes outside of C2. Use Green’s Theorem to evaluate ~ (a) D curl(V ) dA where D denotes the area inside C3 but outside C4; ~ (b) RRD curl(V ) dA where D denotes the area inside C2 but outside C1; ~ (c) RRD curl(V ) dA where D denotes the area inside C2 but outside C4; ~ (d) RRD curl(V ) dA where D denotes the area inside C2 but outside both C1 and C4; (e) RR V~ Tdsˆ . C6 · 4. Let SRr denote the square of side 2r given by:

a r x a + r,   b r y b + r.  

Thus the center of the square is the point (a, b). Let Cr be the boundary of that square, parametrized counter-clockwise. Let V~ (x, y)= P (x, y),Q(x, y) be some vector ﬁeld deﬁned near (a, b). Show that h i

V~ Tdsˆ @Q @P lim Cr · = (a, b) (a, b). r 0 Area(S ) @x @y ! R r Discuss the intuitive meaning of this result in a sentence or so. 5.6. 3D CURL; STOKES’ THEOREM 85

5.6 3D curl; Stokes’ Theorem

Key Ideas.

Geometric idea of 3D curl: The 3D curl (usually just called “curl”) describes the total rotation • of a vector ﬁeld in R3.IfAˆ is any unit vector, then curl(V~ ) Aˆ represents the extent to which V~ · is rotating about the axis deﬁned by Aˆ. (This is equivalent to saying that curl(V~ ) Aˆ represents · the extent to which V~ is rotating in the plane that has Aˆ as its unit normal.) Thus curl(V~ ) points in the direction about which there is the most rotation, which the magnitude equal to the amount of rotation and the sign determined by the “right hand” positivity convention.

Deﬁnition of curl: Suppose V~ = V ,V ,V . Then • h x y zi curl(V~ )= curl( V ,V ), curl( V ,V ), curl( V ,V ). h h y zi h x zi h x yi The minus sign is due to i k = j. ⇥ Computation in Cartesian coordinates: Suppose V~ = V ,V ,V . Then • h x y zi @V @V @V @V @V @V curl(V~ )= z y , z x , y x @y @z @x @z @x @y ⌧ ✓ ◆ Computation using determinants:

@ i @x Vx @ curl(V~ )=det0j @y Vy1 @ Bk @z Vz C B C @ A Di↵erentiation rules: • – Scaling rule: curl(cV~ )=c curl(V~ ) – Addition rule: curl(V~ + W~ )=curl(V~ )+curl(W~ ) – Product rule: curl(fV~ ) = grad(f) V~ + f curl(V~ ). ⇥ The 3D Curl Theorem. Suppose S is a (two-sided) surface in R3 and that the curve C is the • boundary of S. We orient S and C by choosing unit normal vector Nˆ on S and unit tangent vector Tˆ on C so that Tˆ Nˆ is the outward normal to S. Then for any smooth vector ﬁeld V~ we have ⇥ curl(V~ ) NdAˆ = V~ Tds.ˆ · · ZZS ZC The 3D Curl Theorem is called Stokes’ Theorem (though Stokes’ theorem is more general).

Justiﬁcation: Divide D in to many squares . . . • 86 CHAPTER 5. FUNDAMENTAL THEOREMS

Exercises. 1. Compute the curl of the following vector ﬁelds: (a) V~ = x, y, z h i (b) V~ = z,x,y h i (c) V~ = 0,z, y h i (d) V~ = x + y, y + z,z + x h i 2. Let V~ (x, y, z)= yz, xz, xy . h i (a) What is the circulation rate of V~ around the vector Nˆ = 1 1, 1, 1 at the following p3 h i points: i. (1, 0, 0); ii. (0, 1, 0); iii. (0, 0, 1). (b) Repeat the above for the circulation around the vector Nˆ = 0, 1, 0 . h i (c) Around which vector (or in other words, in which plane) does V~ circulate the most? What is the maximum circulation rate?

3. Consider the configuration in Figure 5.2. Note: the surface S1 is just the “volcanos” up to C3, the surface S2 is just the “head” and the surface S3 is just the bowl on the bottom. The normal vector field Nˆ is outward pointing, as indicated. Suppose V~ (x, y, z) is a vector field defined (and smooth) throughout the xyz-space. Use Stokes’ Theorem to evaluate (a) V~ Tdsˆ if it is known that curl(V~ ) NdAˆ = 2; C3 · S3 · (b) R curl(V~ ) NdAˆ if it is knownRR that curl(V~ ) NdAˆ = 1; S2 · S3 · (c) RR curl(V~ ) NdAˆ if it is known that RR V~ Tdsˆ = V~ Tdsˆ = V~ Tdsˆ = 3; S1 · C1 · C2 · C3 · (d) RR V~ Tdsˆ if it is known that curl(R V~ ) NdAˆ R= curl(V~R) NdAˆ = 3 and C2 · S1 · S2 · V~ Tdsˆ = 2. RC1 · RR RR 4. Let V~R(x, y, z)= y, x, 0 ,letS be the dome of the upper unit hemisphere centered at the origin, and let C beh the equatori of the hemisphere in the xy-plane, oriented counter-clockwise. Compute: (a) The outward flux curl(V~ ) NdAˆ using the definition of the flux integral; S · (b) The circulation RRV~ Tdsˆ using the definition of the circulation integral. C · (c) In a sentence orR so, comment on the relationship between the answers to parts (a) and (b), and the Stokes’ Theorem. ~ ˆ 5. Use Stokes’ Theorem to evaluate the circulation integrals C V Tdsfor the following choices ~ · of V and C. R 5.6. 3D CURL; STOKES’ THEOREM 87

Nˆ

C1 C2 S2

Nˆ S1 Nˆ

Nˆ C3

Figure 5.2: The ﬁgure for Problem 3.

(a) V~ (x, y, z)= z,x,y while C is the contour of the triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1)h in thati order. (b) V~ (x, y, z)= x3,y3,z3 while C is the contour of a triangle with vertices (0, 0, 0), (1, 0, 1) and (0, 1, 1)h traversedi in that order. (c) V~ (x, y, z)= 0, z,y while C is the North 45-th parallel of the unit sphere centered at the origin, orientedh counterclockwisei when viewed from the above. 88 CHAPTER 5. FUNDAMENTAL THEOREMS

5.7 The infamous “del” notation

Much of the physics and math literature uses the formal object “del” given by

@ @ @ @ @ @ ~ = , , = i + j + k r @x @y @z @x @y @z ⌧ to concisely express ideas in vector calculus. It is important to keep in mind that while this object uses vector notation, it is not a vector – it’s just a formal object that allows us to write down various formulas in a convenient way. Here is a short summary of facts and formulas that can be written using the del object.

Derivatives (gradient, divergence, curl): • grad(f)=~ f div(V~ )=~ V~ curl(V~ )=~ V~ r r· r⇥ Product rules for gradient, divergence, curl: • ~ (fg)=g~ f + f ~ g ~ (fV~ )=~ f V~ + f ~ V~ ~ (fV~ )=~ f V~ + f ~ V~ r r r r· r · r· r⇥ r ⇥ r⇥ Divergence Theorem: • ~ VdV~ = V~ NdAˆ r· · ZZZD ZZC Curl Theorem: • (~ V~ ) NdAˆ = V~ Tds.ˆ r⇥ · · ZZS ZC The Laplacian of a function has two common notations: • div(grad(u)) = u div(grad(u)) = ~ 2u. r Exercises. 1. Let u be a function. (a) Translate the quantity ~ (~ u) into the language of grad/div/curl. r⇥ r (b) Show by brute force computation that ~ (~ u) = 0 for all functions u. r⇥ r 2. Let V~ be a vector ﬁeld. (a) Translate the quantity div(curl(V~ )) into del notation. (b) Show by brute force computation that div(curl(V~ )) = 0 for all vector ﬁelds V~ .

3. Show by brute force computation that ~ (A~ B~ )= (~ A~) B~ + A~ (~ B~ ). r· ⇥ r⇥ · · r⇥