Math 241: Multivariable Calculus, Lecture 36 Stokes and Divergence Theorem, Section 16.8, 16.9

Math 241: Multivariable calculus, Lecture 36 Stokes and Divergence Theorem, Section 16.8, 16.9

go.illinois.edu/math241fa17

Wednesday, December 6th, 2017

go.illinois.edu/math241fa17. • stand on S with head pointing in direction n, • look at boundary curves, • orient the curves to the left.

S a surface, oriented by n, bounded by ﬁnitely many simple closed curves C1,..., Ck . Then ∂S = C1 ∪ ... ∪ Ck oriented by the rule:

Generalized Stokes Theorem For any continuously diﬀerentiable 3 vector ﬁeld F on R Z ZZ F · dr = curl(F) · dS ∂S S

Generalized Stokes Theorem: multiple boundary components.

• Final exam, December 16th, 1:30pm–4:30pm. Sign up for conﬂicts!

go.illinois.edu/math241fa17. • Final exam, December 16th, 1:30pm–4:30pm. Sign up for conﬂicts!

• look at boundary curves, • orient the curves to the left.

Generalized Stokes Theorem For any continuously diﬀerentiable 3 vector ﬁeld F on R Z ZZ F · dr = curl(F) · dS ∂S S

Generalized Stokes Theorem: multiple boundary components.

S a surface, oriented by n, bounded by ﬁnitely many simple closed curves C1,..., Ck . Then ∂S = C1 ∪ ... ∪ Ck oriented by the rule: • stand on S with head pointing in direction n,

go.illinois.edu/math241fa17. • Final exam, December 16th, 1:30pm–4:30pm. Sign up for conﬂicts!

• orient the curves to the left.

Generalized Stokes Theorem For any continuously diﬀerentiable 3 vector ﬁeld F on R Z ZZ F · dr = curl(F) · dS ∂S S

Generalized Stokes Theorem: multiple boundary components.

S a surface, oriented by n, bounded by ﬁnitely many simple closed curves C1,..., Ck . Then ∂S = C1 ∪ ... ∪ Ck oriented by the rule: • stand on S with head pointing in direction n, • look at boundary curves,

go.illinois.edu/math241fa17. • Final exam, December 16th, 1:30pm–4:30pm. Sign up for conﬂicts!

Generalized Stokes Theorem For any continuously diﬀerentiable 3 vector ﬁeld F on R Z ZZ F · dr = curl(F) · dS ∂S S

Generalized Stokes Theorem: multiple boundary components.

go.illinois.edu/math241fa17. • Final exam, December 16th, 1:30pm–4:30pm. Sign up for conﬂicts!

Generalized Stokes Theorem For any continuously diﬀerentiable 3 vector ﬁeld F on R Z ZZ F · dr = curl(F) · dS ∂S S

Generalized Stokes Theorem: multiple boundary components.

go.illinois.edu/math241fa17. • Final exam, December 16th, 1:30pm–4:30pm. Sign up for conﬂicts!

Generalized Stokes Theorem: multiple boundary components.

Generalized Stokes Theorem For any continuously diﬀerentiable 3 vector ﬁeld F on R Z ZZ F · dr = curl(F) · dS ∂S S

go.illinois.edu/math241fa17. Z Calculate F · dr. C Hint: It’s not zero!

Let C0 be the unit circle in the xy-plane oriented counter- clockwise. Let S be a surface with ∂S = C − C0, which does not intersect {(x, y, z) | x = y = 0}. Then Stokes tells us ZZ Z Z curl(F) · dS = F · dr − F · dr. C C0 S

D y −x E Let F = x2+y 2 , x2+y 2 , z and C be the oriented curve deﬁned by r(t) = h2 cos(t), 3 sin(t), 2 + sin(20t)i, t ∈ [0, 2π]

Example

go.illinois.edu/math241fa17. Hint: It’s not zero!

Example

D y −x E Let F = x2+y 2 , x2+y 2 , z and C be the oriented curve deﬁned by r(t) = h2 cos(t), 3 sin(t), 2 + sin(20t)i, t ∈ [0, 2π] Z Calculate F · dr. C

go.illinois.edu/math241fa17. Let C0 be the unit circle in the xy-plane oriented counter- clockwise. Let S be a surface with ∂S = C − C0, which does not intersect {(x, y, z) | x = y = 0}. Then Stokes tells us ZZ Z Z curl(F) · dS = F · dr − F · dr. C C0 S

Example

D y −x E Let F = x2+y 2 , x2+y 2 , z and C be the oriented curve deﬁned by r(t) = h2 cos(t), 3 sin(t), 2 + sin(20t)i, t ∈ [0, 2π] Z Calculate F · dr. C Hint: It’s not zero!

go.illinois.edu/math241fa17. Then Stokes tells us ZZ Z Z curl(F) · dS = F · dr − F · dr. C C0 S

Example

D y −x E Let F = x2+y 2 , x2+y 2 , z and C be the oriented curve deﬁned by r(t) = h2 cos(t), 3 sin(t), 2 + sin(20t)i, t ∈ [0, 2π] Z Calculate F · dr. C Hint: It’s not zero!

Let C0 be the unit circle in the xy-plane oriented counter- clockwise. Let S be a surface with ∂S = C − C0, which does not intersect {(x, y, z) | x = y = 0}.

go.illinois.edu/math241fa17. Example

D y −x E Let F = x2+y 2 , x2+y 2 , z and C be the oriented curve deﬁned by r(t) = h2 cos(t), 3 sin(t), 2 + sin(20t)i, t ∈ [0, 2π] Z Calculate F · dr. C Hint: It’s not zero!

go.illinois.edu/math241fa17. • First note that curl(F) = 0 so we need only the integral around C0. • C0 is parameterized by r(t) = hcos(t), sin(t), 0i, 0 ≤ t ≤ 2π, so the velocity vector is r0(t) = h− sin(t), cos(t), 0i • F(r(t)) = hsin(t), − cos(t), 0i, • F(r(t)) · r0(t) = hsin(t), − cos(t), 0i · h− sin(t), cos(t), 0i = −1 • Z Z 2π F · dr = (−1) dt = −2π. C0 0

Example Continued

y −x F = , , z x2 + y 2 x2 + y 2 Z Z F · dr = x curl(F) · dS + F · dr. C S C0

go.illinois.edu/math241fa17. • C0 is parameterized by r(t) = hcos(t), sin(t), 0i, 0 ≤ t ≤ 2π, so the velocity vector is r0(t) = h− sin(t), cos(t), 0i • F(r(t)) = hsin(t), − cos(t), 0i, • F(r(t)) · r0(t) = hsin(t), − cos(t), 0i · h− sin(t), cos(t), 0i = −1 • Z Z 2π F · dr = (−1) dt = −2π. C0 0

Example Continued

y −x F = , , z x2 + y 2 x2 + y 2 Z Z F · dr = x curl(F) · dS + F · dr. C S C0

• First note that curl(F) = 0 so we need only the integral around C0.

go.illinois.edu/math241fa17. • F(r(t)) = hsin(t), − cos(t), 0i, • F(r(t)) · r0(t) = hsin(t), − cos(t), 0i · h− sin(t), cos(t), 0i = −1 • Z Z 2π F · dr = (−1) dt = −2π. C0 0

Example Continued

y −x F = , , z x2 + y 2 x2 + y 2 Z Z F · dr = x curl(F) · dS + F · dr. C S C0

go.illinois.edu/math241fa17. • F(r(t)) · r0(t) = hsin(t), − cos(t), 0i · h− sin(t), cos(t), 0i = −1 • Z Z 2π F · dr = (−1) dt = −2π. C0 0

Example Continued

y −x F = , , z x2 + y 2 x2 + y 2 Z Z F · dr = x curl(F) · dS + F · dr. C S C0

go.illinois.edu/math241fa17. • Z Z 2π F · dr = (−1) dt = −2π. C0 0

Example Continued

y −x F = , , z x2 + y 2 x2 + y 2 Z Z F · dr = x curl(F) · dS + F · dr. C S C0

go.illinois.edu/math241fa17. Example Continued

y −x F = , , z x2 + y 2 x2 + y 2 Z Z F · dr = x curl(F) · dS + F · dr. C S C0

• First note that curl(F) = 0 so we need only the integral around C0. • C0 is parameterized by r(t) = hcos(t), sin(t), 0i, 0 ≤ t ≤ 2π, so the velocity vector is r0(t) = h− sin(t), cos(t), 0i • F(r(t)) = hsin(t), − cos(t), 0i, • F(r(t)) · r0(t) = hsin(t), − cos(t), 0i · h− sin(t), cos(t), 0i = −1 • Z Z 2π F · dr = (−1) dt = −2π. C0 0 go.illinois.edu/math241fa17. • Let DR be a disk centered at P, radius R, normal vector n. H • Circulation of ﬂuid around circle ∂DR is F · dr. ∂DR • Rate of circulation per unit area at P around n is 1 H limR→0 F · dr. Area DR ∂DR • By Stokes this is 1 lim x curl(F) · n dS = curl(F(P)) · n. R→0 Area DR DR

• So curl(F(P)) encodes the rate of circulation per unit area around inﬁnitesimal circles.

F = velocity field of fluid flow. Then curl(F) describes infinitesimal rotation of F. To see this use Stokes:

Back to the meaning of curl

go.illinois.edu/math241fa17. • Let DR be a disk centered at P, radius R, normal vector n. H • Circulation of ﬂuid around circle ∂DR is F · dr. ∂DR • Rate of circulation per unit area at P around n is 1 H limR→0 F · dr. Area DR ∂DR • By Stokes this is 1 lim x curl(F) · n dS = curl(F(P)) · n. R→0 Area DR DR

• So curl(F(P)) encodes the rate of circulation per unit area around inﬁnitesimal circles.

Then curl(F) describes inﬁnitesimal rotation of F. To see this use Stokes:

Back to the meaning of curl

F = velocity field of fluid flow.

• So curl(F(P)) encodes the rate of circulation per unit area around inﬁnitesimal circles.

To see this use Stokes:

Back to the meaning of curl

F = velocity field of fluid flow. Then curl(F) describes infinitesimal rotation of F.

• So curl(F(P)) encodes the rate of circulation per unit area around inﬁnitesimal circles.

Back to the meaning of curl

F = velocity field of fluid flow. Then curl(F) describes infinitesimal rotation of F. To see this use Stokes:

go.illinois.edu/math241fa17. H • Circulation of ﬂuid around circle ∂DR is F · dr. ∂DR • Rate of circulation per unit area at P around n is 1 H limR→0 F · dr. Area DR ∂DR • By Stokes this is 1 lim x curl(F) · n dS = curl(F(P)) · n. R→0 Area DR DR

• So curl(F(P)) encodes the rate of circulation per unit area around inﬁnitesimal circles.

Back to the meaning of curl

F = velocity field of fluid flow. Then curl(F) describes infinitesimal rotation of F. To see this use Stokes:

• Let DR be a disk centered at P, radius R, normal vector n.

go.illinois.edu/math241fa17. • Rate of circulation per unit area at P around n is 1 H limR→0 F · dr. Area DR ∂DR • By Stokes this is 1 lim x curl(F) · n dS = curl(F(P)) · n. R→0 Area DR DR

• So curl(F(P)) encodes the rate of circulation per unit area around inﬁnitesimal circles.

Back to the meaning of curl

F = velocity field of fluid flow. Then curl(F) describes infinitesimal rotation of F. To see this use Stokes:

• Let DR be a disk centered at P, radius R, normal vector n. H • Circulation of ﬂuid around circle ∂DR is F · dr. ∂DR

go.illinois.edu/math241fa17. • By Stokes this is 1 lim x curl(F) · n dS = curl(F(P)) · n. R→0 Area DR DR

• So curl(F(P)) encodes the rate of circulation per unit area around inﬁnitesimal circles.

Back to the meaning of curl

F = velocity field of fluid flow. Then curl(F) describes infinitesimal rotation of F. To see this use Stokes:

• Let DR be a disk centered at P, radius R, normal vector n. H • Circulation of ﬂuid around circle ∂DR is F · dr. ∂DR • Rate of circulation per unit area at P around n is 1 H limR→0 F · dr. Area DR ∂DR

go.illinois.edu/math241fa17. • So curl(F(P)) encodes the rate of circulation per unit area around inﬁnitesimal circles.

Back to the meaning of curl

F = velocity field of fluid flow. Then curl(F) describes infinitesimal rotation of F. To see this use Stokes:

go.illinois.edu/math241fa17. Back to the meaning of curl

F = velocity field of fluid flow. Then curl(F) describes infinitesimal rotation of F. To see this use Stokes:

• So curl(F(P)) encodes the rate of circulation per unit area around inﬁnitesimal circles.

go.illinois.edu/math241fa17. 3 A closed surface S is the boundary of a region D in R . For example a sphere x2 + y 2 + y 2 = R2 is the boundary of the ball x2 + y 2 + z2 ≤ R2. The positive orientation for a closed surface S is the outward pointing normal.

Theorem. If S is a closed surface and F is vector ﬁeld with continuous partial derivatives, then

x curl(F) · dS = 0 S Why? The boundary of a boundary is empty: ∂S = ∅, so curl(F) · dS = R F · dr = 0. sS ∂S

Another special case of Stokes Theorem

go.illinois.edu/math241fa17. For example a sphere x2 + y 2 + y 2 = R2 is the boundary of the ball x2 + y 2 + z2 ≤ R2. The positive orientation for a closed surface S is the outward pointing normal.

Theorem. If S is a closed surface and F is vector ﬁeld with continuous partial derivatives, then

x curl(F) · dS = 0 S Why? The boundary of a boundary is empty: ∂S = ∅, so curl(F) · dS = R F · dr = 0. sS ∂S

Another special case of Stokes Theorem

3 A closed surface S is the boundary of a region D in R .

go.illinois.edu/math241fa17. The positive orientation for a closed surface S is the outward pointing normal.

Theorem. If S is a closed surface and F is vector ﬁeld with continuous partial derivatives, then

x curl(F) · dS = 0 S Why? The boundary of a boundary is empty: ∂S = ∅, so curl(F) · dS = R F · dr = 0. sS ∂S

Another special case of Stokes Theorem

3 A closed surface S is the boundary of a region D in R . For example a sphere x2 + y 2 + y 2 = R2 is the boundary of the ball x2 + y 2 + z2 ≤ R2.

go.illinois.edu/math241fa17. Theorem. If S is a closed surface and F is vector ﬁeld with continuous partial derivatives, then

x curl(F) · dS = 0 S Why? The boundary of a boundary is empty: ∂S = ∅, so curl(F) · dS = R F · dr = 0. sS ∂S

Another special case of Stokes Theorem

3 A closed surface S is the boundary of a region D in R . For example a sphere x2 + y 2 + y 2 = R2 is the boundary of the ball x2 + y 2 + z2 ≤ R2. The positive orientation for a closed surface S is the outward pointing normal.

go.illinois.edu/math241fa17. Why? The boundary of a boundary is empty: ∂S = ∅, so curl(F) · dS = R F · dr = 0. sS ∂S

Another special case of Stokes Theorem

Theorem. If S is a closed surface and F is vector ﬁeld with continuous partial derivatives, then

x curl(F) · dS = 0 S

go.illinois.edu/math241fa17. The boundary of a boundary is empty: ∂S = ∅, so curl(F) · dS = R F · dr = 0. sS ∂S

Another special case of Stokes Theorem

Theorem. If S is a closed surface and F is vector ﬁeld with continuous partial derivatives, then

x curl(F) · dS = 0 S Why?

go.illinois.edu/math241fa17. so curl(F) · dS = R F · dr = 0. sS ∂S

Another special case of Stokes Theorem

Theorem. If S is a closed surface and F is vector ﬁeld with continuous partial derivatives, then

x curl(F) · dS = 0 S Why? The boundary of a boundary is empty: ∂S = ∅,

go.illinois.edu/math241fa17. Another special case of Stokes Theorem

Theorem. If S is a closed surface and F is vector ﬁeld with continuous partial derivatives, then

x curl(F) · dS = 0 S Why? The boundary of a boundary is empty: ∂S = ∅, so curl(F) · dS = R F · dr = 0. sS ∂S

go.illinois.edu/math241fa17. H H • ∂S = C0 − C1. Then curl(F) · dS + F · dr = F · dr sS C1 C0 • If ∂S = C − C and curl(F) = 0, then H F · dr = H F · dr 0 1 C1 C0 • ∂S1 = ∂S2. Then curl(F) · dS = curl(F) · dS sS1 sS2 • S a closed surface. Then curl(F) · dS = 0. sS

Let F be a vector ﬁeld (with continuous partial derivatives; “nice”).

Summary Special Cases Stokes Theorem.

Let F be a vector ﬁeld (with continuous partial derivatives; “nice”).

go.illinois.edu/math241fa17. • If ∂S = C − C and curl(F) = 0, then H F · dr = H F · dr 0 1 C1 C0 • ∂S1 = ∂S2. Then curl(F) · dS = curl(F) · dS sS1 sS2 • S a closed surface. Then curl(F) · dS = 0. sS

Summary Special Cases Stokes Theorem.

Let F be a vector ﬁeld (with continuous partial derivatives; “nice”). H H • ∂S = C0 − C1. Then curl(F) · dS + F · dr = F · dr sS C1 C0

go.illinois.edu/math241fa17. • ∂S1 = ∂S2. Then curl(F) · dS = curl(F) · dS sS1 sS2 • S a closed surface. Then curl(F) · dS = 0. sS

Summary Special Cases Stokes Theorem.

Let F be a vector ﬁeld (with continuous partial derivatives; “nice”). H H • ∂S = C0 − C1. Then curl(F) · dS + F · dr = F · dr sS C1 C0 • If ∂S = C − C and curl(F) = 0, then H F · dr = H F · dr 0 1 C1 C0

go.illinois.edu/math241fa17. • S a closed surface. Then curl(F) · dS = 0. sS

Summary Special Cases Stokes Theorem.

Let F be a vector ﬁeld (with continuous partial derivatives; “nice”). H H • ∂S = C0 − C1. Then curl(F) · dS + F · dr = F · dr sS C1 C0 • If ∂S = C − C and curl(F) = 0, then H F · dr = H F · dr 0 1 C1 C0 • ∂S1 = ∂S2. Then curl(F) · dS = curl(F) · dS sS1 sS2

go.illinois.edu/math241fa17. Summary Special Cases Stokes Theorem.

go.illinois.edu/math241fa17. Idea: Stokes gives relation between 2 dim integral (of curlF) and the 1 dim integral (of F). What about 3 and 2 dimensional integrals? 3 Theorem. If D is a region in R bounded by a surface S, given the positive orientation, and F is a vector ﬁeld with continuous partial derivatives, then

y div(F) dV = x F · dS. D S We sometimes write ∂D = S for the oriented boundary. Then this becomes y div(F) dV = x F · dS. D ∂D

Since div(curl(F)) = 0, the previous special case of Stokes Theorem is also a special case of the Divergence Theorem.

The Divergence Theorem

go.illinois.edu/math241fa17. What about 3 and 2 dimensional integrals? 3 Theorem. If D is a region in R bounded by a surface S, given the positive orientation, and F is a vector ﬁeld with continuous partial derivatives, then

y div(F) dV = x F · dS. D S We sometimes write ∂D = S for the oriented boundary. Then this becomes y div(F) dV = x F · dS. D ∂D

Since div(curl(F)) = 0, the previous special case of Stokes Theorem is also a special case of the Divergence Theorem.

The Divergence Theorem

Idea: Stokes gives relation between 2 dim integral (of curlF) and the 1 dim integral (of F).

go.illinois.edu/math241fa17. 3 Theorem. If D is a region in R bounded by a surface S, given the positive orientation, and F is a vector ﬁeld with continuous partial derivatives, then

y div(F) dV = x F · dS. D S We sometimes write ∂D = S for the oriented boundary. Then this becomes y div(F) dV = x F · dS. D ∂D

Since div(curl(F)) = 0, the previous special case of Stokes Theorem is also a special case of the Divergence Theorem.

The Divergence Theorem

Idea: Stokes gives relation between 2 dim integral (of curlF) and the 1 dim integral (of F). What about 3 and 2 dimensional integrals?

go.illinois.edu/math241fa17. We sometimes write ∂D = S for the oriented boundary. Then this becomes y div(F) dV = x F · dS. D ∂D

Since div(curl(F)) = 0, the previous special case of Stokes Theorem is also a special case of the Divergence Theorem.

The Divergence Theorem

Idea: Stokes gives relation between 2 dim integral (of curlF) and the 1 dim integral (of F). What about 3 and 2 dimensional integrals? 3 Theorem. If D is a region in R bounded by a surface S, given the positive orientation, and F is a vector ﬁeld with continuous partial derivatives, then

y div(F) dV = x F · dS. D S

go.illinois.edu/math241fa17. Then this becomes y div(F) dV = x F · dS. D ∂D

Since div(curl(F)) = 0, the previous special case of Stokes Theorem is also a special case of the Divergence Theorem.

The Divergence Theorem

y div(F) dV = x F · dS. D S We sometimes write ∂D = S for the oriented boundary.

go.illinois.edu/math241fa17. Since div(curl(F)) = 0, the previous special case of Stokes Theorem is also a special case of the Divergence Theorem.

The Divergence Theorem

y div(F) dV = x F · dS. D S We sometimes write ∂D = S for the oriented boundary. Then this becomes y div(F) dV = x F · dS. D ∂D

go.illinois.edu/math241fa17. The Divergence Theorem

y div(F) dV = x F · dS. D S We sometimes write ∂D = S for the oriented boundary. Then this becomes y div(F) dV = x F · dS. D ∂D

Since div(curl(F)) = 0, the previous special case of Stokes Theorem is also a special case of the Divergence Theorem. go.illinois.edu/math241fa17. Example 1. Let S be the unit sphere with the outward pointing normal. Calculate

y 2 x hcos(z) + e , x + y, z + 3xi · dS S

Example 2. Let S be the part of the cone z2 = x2 + y 2 for 0 ≤ z ≤ 2, with outward pointing normal. Calculate the ﬂux over the cone of the vector ﬁeld F = hz + cosh(y), sin(z2), 2i

Examples

go.illinois.edu/math241fa17. Example 2. Let S be the part of the cone z2 = x2 + y 2 for 0 ≤ z ≤ 2, with outward pointing normal. Calculate the ﬂux over the cone of the vector ﬁeld F = hz + cosh(y), sin(z2), 2i

Examples

Example 1. Let S be the unit sphere with the outward pointing normal. Calculate

y 2 x hcos(z) + e , x + y, z + 3xi · dS S

go.illinois.edu/math241fa17. Examples

Example 1. Let S be the unit sphere with the outward pointing normal. Calculate

y 2 x hcos(z) + e , x + y, z + 3xi · dS S

Example 2. Let S be the part of the cone z2 = x2 + y 2 for 0 ≤ z ≤ 2, with outward pointing normal. Calculate the ﬂux over the cone of the vector ﬁeld F = hz + cosh(y), sin(z2), 2i

go.illinois.edu/math241fa17. BR = ball of radius R > 0 centered at a point P.

xF · dS = net rate ﬂuid ﬂows through the sphere ∂BR . ∂BR

= net rate of volume expansion of ﬂuid in BR .

rate of volume 1 expansion per unit = lim x F · dS R→0 vol(BR ) volume at P ∂BR

1 = lim y div(F) dV = div(F)(P) R→0 vol(BR ) BR So the divergence of F measures the rate of volume expansion.

Recall that if F is the velocity field of a fluid flow, then the flux of F over a surface S is the net rate of fluid flowing past S.

The meaning of divergence

go.illinois.edu/math241fa17. xF · dS = net rate ﬂuid ﬂows through the sphere ∂BR . ∂BR

= net rate of volume expansion of ﬂuid in BR .

rate of volume 1 expansion per unit = lim x F · dS R→0 vol(BR ) volume at P ∂BR

1 = lim y div(F) dV = div(F)(P) R→0 vol(BR ) BR So the divergence of F measures the rate of volume expansion.

The meaning of divergence

Recall that if F is the velocity field of a fluid flow, then the flux of F over a surface S is the net rate of fluid flowing past S. BR = ball of radius R > 0 centered at a point P.

go.illinois.edu/math241fa17. = net rate of volume expansion of ﬂuid in BR .

rate of volume 1 expansion per unit = lim x F · dS R→0 vol(BR ) volume at P ∂BR

1 = lim y div(F) dV = div(F)(P) R→0 vol(BR ) BR So the divergence of F measures the rate of volume expansion.

The meaning of divergence

Recall that if F is the velocity field of a fluid flow, then the flux of F over a surface S is the net rate of fluid flowing past S. BR = ball of radius R > 0 centered at a point P. xF · dS = net rate fluid flows through the sphere ∂BR . ∂BR

go.illinois.edu/math241fa17. rate of volume 1 expansion per unit = lim x F · dS R→0 vol(BR ) volume at P ∂BR

1 = lim y div(F) dV = div(F)(P) R→0 vol(BR ) BR So the divergence of F measures the rate of volume expansion.

The meaning of divergence

= net rate of volume expansion of ﬂuid in BR .

go.illinois.edu/math241fa17. 1 = lim y div(F) dV = div(F)(P) R→0 vol(BR ) BR So the divergence of F measures the rate of volume expansion.

The meaning of divergence

= net rate of volume expansion of ﬂuid in BR .

rate of volume 1 expansion per unit = lim x F · dS R→0 vol(BR ) volume at P ∂BR

go.illinois.edu/math241fa17. = div(F)(P)

So the divergence of F measures the rate of volume expansion.

The meaning of divergence

= net rate of volume expansion of ﬂuid in BR .

rate of volume 1 expansion per unit = lim x F · dS R→0 vol(BR ) volume at P ∂BR

1 = lim y div(F) dV . R→0 vol(BR ) BR

go.illinois.edu/math241fa17. The meaning of divergence

= net rate of volume expansion of ﬂuid in BR .

rate of volume 1 expansion per unit = lim x F · dS R→0 vol(BR ) volume at P ∂BR

1 = lim y div(F) dV = div(F)(P). R→0 vol(BR ) BR So the divergence of F measures the rate of volume expansion.

go.illinois.edu/math241fa17. Extra Practice problem

Problem: Calculate x curl(F) · dS, y S where S is the oriented surface shown, x 2 y 2 bounded by the ellipse x + 4 = 1 in the xy-plane, and F = hy, −x, zi

go.illinois.edu/math241fa17. • No class Friday, Monday and Wednesday • Final exam, December 16th, 1:30pm–4:30pm. Sign up for conﬂicts! • Good luck for the ﬁnal!

The End

That is it!

go.illinois.edu/math241fa17. • Final exam, December 16th, 1:30pm–4:30pm. Sign up for conﬂicts! • Good luck for the ﬁnal!

The End

That is it! • No class Friday, Monday and Wednesday

go.illinois.edu/math241fa17. • Good luck for the ﬁnal!

The End

That is it! • No class Friday, Monday and Wednesday • Final exam, December 16th, 1:30pm–4:30pm. Sign up for conﬂicts!

go.illinois.edu/math241fa17. The End

That is it! • No class Friday, Monday and Wednesday • Final exam, December 16th, 1:30pm–4:30pm. Sign up for conﬂicts! • Good luck for the ﬁnal!

go.illinois.edu/math241fa17.