Calculus

Table of Contents

Calculus

Definitions

Norm / mesh of partition

The norm ( or mesh ) of the partition

calculus_cbaec12fac8803e211fb681ccb58784e2968899d.png

is the length of the longest of these subintervals:

calculus_77cd90405b37487cb3735d44c49222b44d3bac82.png

Riemann-Stieltjes integral

The Riemann-Stieltjes integral of a real-valued function calculus_93369077affb352dbdb97e8b3182fd50784f2b14.png of a real variable with respect to a real function calculus_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png is denoted by

calculus_04fac3399f89f9ea2a5d7e1fc184aea11ba471d9.png

and defined to be the limit, as the norm of the partition

calculus_96f065eea3b866ed9af800982f0f287e53a7cfaa.png

of the interval calculus_b2fdf0617d736136c4564c3795fed8545c0717bf.png approaches zero, of the approximating sum

calculus_e3eff46e8da46f41919ab05e6b5a192644567a39.png

where calculus_922c1b6b841061e4343cd3f25d74e4166e89b9ec.png is in the i-th subinterval calculus_50ad8efe663097830cd636f611b79c365d190945.png. The two functions calculus_93369077affb352dbdb97e8b3182fd50784f2b14.png and calculus_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png are respectively called the integrand and the integrator.

The "limit" is understood to be a number calculus_e46729bc781c25bbc7120ee2892cc1c0215af7da.png (value of the Riemann-Stieltjes integral) such that for every calculus_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png there exists a calculus_0809b466cd9de765ec30e362bd8646103cb497eb.png s.t. every partition calculus_a9090c77ce9916955c745920bf8f134a0932d59d.png with calculus_1c01dc9d1a22dcefdd5500b7a94b9d337060149c.png, and for every choice of points calculus_922c1b6b841061e4343cd3f25d74e4166e89b9ec.png in calculus_50ad8efe663097830cd636f611b79c365d190945.png,

calculus_3a8878991fbbb59b0cc442737caf09c41448e867.png

Comparison to Riemann integral

The Riemann integral for the case used in the definition above we have

calculus_fcec612edc950471ad94a5ca3fac41762a97eae9.png

Now, what's the difference between this and the Riemann-Stieltjes integral? The main point is that the Riemann-Stieltjes integral allow us to integrate wrt. the function calculus_0264d457a79bf4db510b9d548fc9bd6f44d19bec.png itself rather than to calculus_ddfea20dd1ba24439d6b983e7626c26962cc15ea.png and then wrt. calculus_7a84c9a383f9772338016d101ccc096be06af784.png. In this way it extends the Riemann integral and is slightly more "general".

Equations / Theorems

Fundamental Theorem of Calculus

Jensen's Inequality

Let calculus_9e9944aef408439bfabc565181dd15a4308f5b57.png be a probability space, i.e. calculus_9ccee2f596917251bf95483bcaad79502b5feeb5.png, where

Then if

  • calculus_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png is real-valued function that is calculus_acb0106122b7f90d9bd5639367a141a7e53d8327.png -integrable
  • calculus_078b85cd3478400338e3a1ee425c2a468644be7e.png is a convex function on the real line

we have:

calculus_8a14a586b2484b9c550af0ed9d81d02874cb251f.png

TODO Proof

Vector calculus

Definitions

Conservative vector field

A conservative vector field is a vector field that is the gradient of some function, know in this context as a scalar potential.

Conservative vector fields have the property that a line integral is path independent , i.e. the choice of any path between two points does not change the value of the line integral.

Equations / Theorems

(Gauss') Divergence Theorem

"Integral definition"

calculus_a467aa84a391387000dd1b515d410b1cdfd8dfb5.png

From what I can tell, this looks like the "approximation" near some point which can be arrived at from the Divergence Theorem using Stokes' Theorem. See this section for what I'm talking about.

Multivariate Taylor Expansion

calculus_1e1934542fbb12d39c98b24bf6f2ea271dcf4c73.png

Vector field identities

calculus_88782ddfd5b73f45bd5bdb1f6ac9be4f0f70e822.png

calculus_53649cab9cb8a5814d58e2cb8ce522472daac817.png

calculus_e7c3058cbfc1b6d732d16156d54cad61a93e275f.png

calculus_6d5c003c5924762a73fa5414590bc34c339bde1f.png

calculus_1dedd49fd002bab6d4a268851f2ca402d5dc1f48.png

calculus_6a228b4adfaefecce19c9729b309cf6c13052539.png

calculus_bf914990ce107fc2cbc74068f216d3f3670295ee.png

calculus_1db0921b83b7b61ecd4d5f8894c98b4d7bface5e.png

calculus_da12c9d2d97a4a57c6ad9ff92d12fbf479f2a385.png

calculus_efa67527c597f22e7f1b3f611b55188f4a7bfb34.png

Elementary identities

calculus_ded7d4fbea7da588508376fcb6fc8cde16d128a1.png

calculus_28724436a017ea01da88be39d10a82d50a813d0b.png

calculus_35aa886e73a98af3897e33602ab3dbe17bc66469.png

calculus_79c741a455b59b0e35781bc26f8be1ae0e1f6e36.png

calculus_5983eb84926363754dcceaaf6aa844d6c48db9f8.png

calculus_2fc1e0ce1031bad3a6b58441f4d7cc6f7ea8e0c7.png

calculus_3af3ceab53bf5e04f0ead6f90f851373ef991a34.png

Essential Calculus: Early Transcendentals

11 Vector Calculus

11.6 Directional Derivatives and the Gradient Vector

Directional Derivative

Finding the derivative of some function calculus_93369077affb352dbdb97e8b3182fd50784f2b14.png at some point calculus_bae4bae48eb3ea91a3404b3c1514d97719620616.png in the direction of some arbitrary unit vector calculus_b30b3586cd29753064cdaf9a5779c7079fb03447.png.

calculus_88a1a2445289c620a19344f9f231de077f28ed69.png

Which expresses the directional derivative in the direction of calculus_b30b3586cd29753064cdaf9a5779c7079fb03447.png as the scalar projection of the gradient vector onto calculus_b30b3586cd29753064cdaf9a5779c7079fb03447.png.

  • Maximizing the Directional Derivative

    Suppose calculus_93369077affb352dbdb97e8b3182fd50784f2b14.png is a differentiable function of two or three variables. The maximum value of the directional derivative calculus_934e318cc560c6e623c52bc3a58cc408a5058096.png is calculus_875694a6439bb454409166b0e968f2c7d57468d5.png and it occurs when calculus_b30b3586cd29753064cdaf9a5779c7079fb03447.png has the same direction as the gradient vector calculus_feb54815374ca4ea9bd36fd81678919a615844c9.png.

    calculus_f0f0a1cb0c113b20bfd69008dcc9b6c70c2e745f.png

    Which is simply maximized when calculus_d63b8463a6a4038909df781dadf8d1ecac8a6af7.png, i.e. calculus_b30b3586cd29753064cdaf9a5779c7079fb03447.png has the same direction as calculus_cf309702f8106a79b659e3ee3517ac9a1eca9a72.png.

    This just says that calculus_93369077affb352dbdb97e8b3182fd50784f2b14.png has it's maximum change in the direction of calculus_cf309702f8106a79b659e3ee3517ac9a1eca9a72.png. If you're at some point calculus_a9090c77ce9916955c745920bf8f134a0932d59d.png, and what the direction to move for maximum change in calculus_93369077affb352dbdb97e8b3182fd50784f2b14.png, move in the direction of calculus_cf309702f8106a79b659e3ee3517ac9a1eca9a72.png.

Tangent Planes to Level Surfaces

Let:

  • calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is a surface with equation calculus_4cdc14d5fc79fbf7ef87d418719426cb1fe205d8.png, i.e. it is a level surface of a function calculus_b2db59aeb94cbc1050079892ff07b21b493513b7.png
  • calculus_26c21c1c3a1ad56fc9f864d32e5a0ce48ef345cd.png be a point on calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png
  • calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png be any curve that lies on the surface calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png and pass through the point calculus_a9090c77ce9916955c745920bf8f134a0932d59d.png, i.e. calculus_259e98fe46a7ea470fc0f3f6f5fb2da87036f9b7.png where calculus_5981551937c9d5a2b4477f3956e0a6eac8a1fe5d.png

Then any point calculus_e6a652bcb10c2dbde5fb97325a7034268247bc4c.png must satisfy the equation of calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png, that is:

calculus_a768cfb66f325b08da544f979c49030b792ad8f9.png

Which says that the gradient vector calculus_939e68c799a059d48c972ca2b3514bd1f202af76.png of calculus_b2db59aeb94cbc1050079892ff07b21b493513b7.png is orthogonal to the gradient of any curve calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png lying on calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png that passes through calculus_a9090c77ce9916955c745920bf8f134a0932d59d.png. Furthermore, this defines the a plane with the normal vector calculus_939e68c799a059d48c972ca2b3514bd1f202af76.png and so we have a rather general definition of a tangent plane.

This makes sense intuitively, because as we move away from the point calculus_a9090c77ce9916955c745920bf8f134a0932d59d.png while staying on the level surface calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png, the value of calculus_b2db59aeb94cbc1050079892ff07b21b493513b7.png does not change. That's the entire point of a level surface: calculus_de588fb0ebf1922efc53aadec76ba85112935a00.png for some calculus_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png. And the direction of maximum increase is then orthogonal to the level surface calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png.

Notice where this comes in? Gradient descent. If we instead move the opposite of the gradient, i.e. orthogonal to the level surface calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png but in the "negative" direction, we got ourselves the entire premise of GD.

11.8 Lagrange Multipliers

Overview

Goal is to find extreme values of calculus_4f0f0ea7a4a2141c6336bcc3049867801f4a1550.png subject to the constraint calculus_a3d6dcbdbfbf1eddee8aca0ac834c61b99a5e1ed.png. In other words, we seek the extreme values of calculus_4f0f0ea7a4a2141c6336bcc3049867801f4a1550.png when the point calculus_40fb973cd997849a029e392c385d32d4b8c40196.png is restriced to lie on the level curve calculus_a3d6dcbdbfbf1eddee8aca0ac834c61b99a5e1ed.png.

To maximize calculus_4f0f0ea7a4a2141c6336bcc3049867801f4a1550.png wrt. calculus_078eeeb369adaed4eb9d1765f75c677ed1df8795.png is then to find the largest value calculus_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png such that the level curve calculus_8e5455a6de51ee85a0712064182c457917400a78.png intersects calculus_a3d6dcbdbfbf1eddee8aca0ac834c61b99a5e1ed.png. This happens when calculus_93369077affb352dbdb97e8b3182fd50784f2b14.png and calculus_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png have the same tangent line . Otherwise, the value of calculus_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png could be increased further. This means that the normal lines at the point calculus_bae4bae48eb3ea91a3404b3c1514d97719620616.png are parallel !

calculus_0ff73e69c634aa1a4c5c4fa13a70fd88513d08dc.png

Algorithm

To find the maximum and minimum values of calculus_4f0f0ea7a4a2141c6336bcc3049867801f4a1550.png subject to the constraint calculus_a3d6dcbdbfbf1eddee8aca0ac834c61b99a5e1ed.png [assuming that these extreme values exist and calculus_3557c07dad9c864306fe62a0a8e8d69a2c2d1dc7.png on the surface calculus_a3d6dcbdbfbf1eddee8aca0ac834c61b99a5e1ed.png]:

a) Find all values of calculus_40fb973cd997849a029e392c385d32d4b8c40196.png and calculus_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png, s.t.:

calculus_e077cb541a7c4b8464c4f7c88c5ff4da30047b7e.png

b) Evaluate calculus_93369077affb352dbdb97e8b3182fd50784f2b14.png at all points calculus_40fb973cd997849a029e392c385d32d4b8c40196.png that result from step (a). The largest of these values is the maximum value of calculus_93369077affb352dbdb97e8b3182fd50784f2b14.png: the smallest is the minimum value of calculus_93369077affb352dbdb97e8b3182fd50784f2b14.png.

13 Vector Calculus

Notation

  • calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png denotes a curve
  • calculus_f5e2d4346d5617845a1b2258ca08e0eccd882968.png defines the path of the curve calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png
  • calculus_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png denotes the region enclosed by the curve calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png
  • positive orientation of a simple closed curve calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png refers to a single counterclockwise traversal of calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png, i.e. the region calculus_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png is always on the left as the point calculus_f5e2d4346d5617845a1b2258ca08e0eccd882968.png traverses calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png

13.4 Green's Theorem

This theorem gives us a neat "trick" to compute integrals for certain types of closed surfaces.

Let calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png be a positively oriented, piecewise-smooth, single closed curve in the plane and let calculus_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png be the region bounded by calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png. If calculus_a9090c77ce9916955c745920bf8f134a0932d59d.png and calculus_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png have continuous partial derivatives on an open region that contains calculus_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png, then

calculus_ba909964410d6b59a09d2f63f1420e1154a69100.png

Notice that we have a proof if we can show that

calculus_5e544aa95c82d58b466f43d7d674257c1b83c704.png

and

calculus_1801f254f5ef8f3882bb1dc05a58c206f923ed2f.png

We write the region calculus_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png as:

calculus_e7102a7a9c8715122e2ee53250e562fe68e7c9ae.png

where calculus_62dfc834e046febca42e35b21e939f37b7e8817d.png and calculus_d5cecca75d6f76ab24a6b55f8ebdc1025cd242cb.png are continuous functions.

Then,

calculus_484e29bcb85e60b07538269fb1f78cd4caa53a81.png

where the last step follows from the Fundamental Theorem of Calculus.

If the entire curve is smooth, we got our result. If it's only piecewise smooth, we then treat the curve calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png as a union of piecewise smooth curves, and compute the integral for each of these piecewise curves making up calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png.

Note that when computing this for the piecewise case, we need to make sure that we choose the correct sign for each of the integrals, i.e. sign s.t. that positive means enclosed surface is on the left.

Green's Theorem can also be extended to include surfaces with holes , i.e. regions which are NOT simply-connected .

This is done by splitting regions in such a way that we instead end up with multiple simply-connected surfaces.

See p. 787 in the book.

Vector forms

The following expresses Green's Theorem as the line integral of the tangential component of calculus_1f15588d4acc55769d76ba21967dd6a42b708989.png along calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png as the double integral of the vertical component of calculus_9c57fc62582f9b1d910629d588d90d503e62cb78.png over the region D enclosed by calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png .

calculus_6eb7ea5ae19667451bea301c746466551bff4df1.png

The other following expresses Green's Theorem as the line integral of the normal component of calculus_1f15588d4acc55769d76ba21967dd6a42b708989.png along calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png is equal to the double integral of the divergence of calculus_1f15588d4acc55769d76ba21967dd6a42b708989.png over the region calculus_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png enclosed by calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png .

calculus_d9b3f6c1d0326924801188c82b5babb6a0cbb7a3.png

13.5 Curl and Divergence

Curl

If calculus_eb9266e4a8fa4e858f80a6f77f52fed1b7c3c450.png is a vector field on calculus_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png and the partial derivatives of calculus_a9090c77ce9916955c745920bf8f134a0932d59d.png, calculus_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png, and calculus_3d136f0fc4860468633907421c098b9feb0eef24.png all exist, then the curl of calculus_1f15588d4acc55769d76ba21967dd6a42b708989.png is the vector field on calculus_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png defined by

calculus_fc2068f6f00dadcf437749a366129cb9a119aa9e.png

  • Why the name "curl"?
    • Curl vecto is associated with rotations
    • Another occus when calculus_1f15588d4acc55769d76ba21967dd6a42b708989.png represents the velocity field in fluid flow. Particles near calculus_d3766507e0688bb06d7d9047eb0d773c07275c28.png in the fluid tend to rotate about the axis that points in the direction of calculus_2b086a667c626313e2d0ae91d2cab7a23c561243.png and the length of this curl vector is a measure of how quickly the particles move around the axis.
Divergence

If calculus_eb9266e4a8fa4e858f80a6f77f52fed1b7c3c450.png is a vector field on calculus_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png and calculus_a9090c77ce9916955c745920bf8f134a0932d59d.png, calculus_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png, and calculus_3d136f0fc4860468633907421c098b9feb0eef24.png have continuous second-order partial derivatives, then

calculus_9fd949e26d9331b0d0a0c6397731ea2f4261eade.png

  • Why the name "divergence"?

    Again, in the context of fluid flow .

    If calculus_64af438d8ff0278191cfdccabace03891d688728.png is the velocity field of a fluid, then calculus_6736fe9dd806fd6ea675886d9ac107b44c05737d.png represents the net rate of change (wrt. time) of the mass of the fluid flowing from point calculus_d3766507e0688bb06d7d9047eb0d773c07275c28.png per unit volume.

    In other words, calculus_a9306b66ccb56dae675b865eb597cd040469ef3f.png measures the tendency of the fluid to diverge from the point calculus_d3766507e0688bb06d7d9047eb0d773c07275c28.png.

    If calculus_0e274ede8b8661eb2b6834903775c14d90360d14.png, then calculus_1f15588d4acc55769d76ba21967dd6a42b708989.png is said to be incompressible.

13.7 Surface Integrals

If calculus_1f15588d4acc55769d76ba21967dd6a42b708989.png is a continuous vector field defined on an oriented surface calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png with a unit normal vector calculus_958858de0fe3ae32a8ace687cda0721254307280.png, then the surface integral of calculus_1f15588d4acc55769d76ba21967dd6a42b708989.png over calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is

calculus_7d077e65a7e90e4591e73a35e3091af1eae15047.png

This integral is also called the flux of calculus_1f15588d4acc55769d76ba21967dd6a42b708989.png across calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png.

And since:

calculus_228b2877c6493c8350c3288e21e899d17b4f77b2.png

we can also a surface integral:

calculus_a63df01542466667b98594900816892540c886bd.png

Applications in Physics
  • Notation
    • calculus_f4d998527bc1b92c69b3a1e28ce712791cfd2edf.png fluid density at calculus_d3766507e0688bb06d7d9047eb0d773c07275c28.png
    • calculus_617b18b3c55689bad1f3306f39ee19706689807b.png velocity field
    • calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is surface the fluid is flowing through
  • Fluid flow

    We can approximate the mass of fluid crossing a segment of the surface, calculus_a2109459542fc0bfaf959cbbbfd07f81ebcf4722.png, in the direction of the normal calculus_958858de0fe3ae32a8ace687cda0721254307280.png per unit time by the quantity:

    calculus_3f11b2ef8f0171a281a66a31509c5a8fd819fd15.png

    where calculus_78d0b09ae53fb5cb18f5d8b060ffafb2b7b91aec.png is the area of the surface-segment calculus_a2109459542fc0bfaf959cbbbfd07f81ebcf4722.png.

    Then by definition of a surface integral, summing these elements and taking the limit we get the rate of flow through the surface calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png as:

    calculus_0fc700f633174b47cb02af8eacce20c9cad99d38.png

    Also known as the flux.

Stoke's Theorem

Let calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png be an oriented piecewise-smooth surface that is bounded by a simple, closed, piecewise-smooth boundary curve calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png with positive orientation. Let calculus_1f15588d4acc55769d76ba21967dd6a42b708989.png be a vector field whose components have continuous partial derivatives on an open region in calculus_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png that contains calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png. Then

calculus_7ac33823838d0649e907f90e42baf06cd5f6de6c.png

The book says I'm not smart enough to know yet. Some day…

Green's Theorem is a special case of Stoke's Theorem.

Shedding light on the curl of a vector field

Using Stoke's Theorem we can shed some more light on calculus_9c57fc62582f9b1d910629d588d90d503e62cb78.png.

Suppose that calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png is an oriented closed curve and calculus_a0bad3442c91c78dc0a25fbf9bcd73061c0b1092.png represents the velocity field in fluid flow. Consider the line integral

calculus_558710a988a5f737d11103fa8a829556300d92e2.png

and recall that calculus_83b0907277086bf223678f075d1bb2e104e466ef.png is the component of calculus_a0bad3442c91c78dc0a25fbf9bcd73061c0b1092.png in the direction of the unit tangent vector calculus_fc9b3740d8cab303a8debd959c24a0b239392d4e.png.

This means that the closer the direction of calculus_a0bad3442c91c78dc0a25fbf9bcd73061c0b1092.png is to the direction of calculus_fc9b3740d8cab303a8debd959c24a0b239392d4e.png, the larger the value of calculus_83b0907277086bf223678f075d1bb2e104e466ef.png. Thus calculus_8476691f653771fbf672cc712b6b621161be6ae3.png is a measure of the tendency of the fluid to move around calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png and is called the circulation of calculus_a0bad3442c91c78dc0a25fbf9bcd73061c0b1092.png around calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png.

Now let calculus_968c12bc9a286a7386421251aff7889348e6727e.png be a point in the fluid and let calculus_9a3bec4a71c60dc26a926c2eda0d24a49bc3269b.png be a small disk with radius calculus_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png and center calculus_66682f4d438b03f5645e94216ad419bff53116d0.png. Then calculus_2f5473678a44a442825ea7c54aee9f864b087c75.png for all points calculus_a9090c77ce9916955c745920bf8f134a0932d59d.png on calculus_9a3bec4a71c60dc26a926c2eda0d24a49bc3269b.png becuase calculus_9c57fc62582f9b1d910629d588d90d503e62cb78.png is continuous.

Thus, by Stokes' Theorem, we get the following approximation to the circulation around the boundary circle calculus_e50495192d7e6248f4820d13fb2219df123de6ba.png:

calculus_352885c6d97fa5609e4af348a561e2392bae0d6c.png

This approximation becomes better as calculus_7ace398a93cfe38d3fd4a79264a3e89e0350ee4c.png and we have

calculus_cc1aa17a4c14e184f2f40c0f326454538c97498a.png

Hence, we have a relationship between the curl and the circulation, where we can view the calculus_e841f88815bdbdb687aaa838c33dd6959fb829f6.png as a measure of the rotating effect of the fluid about the axis calculus_958858de0fe3ae32a8ace687cda0721254307280.png.

The curling effect is greatest about the axis parallel to calculus_86b8563877962201282df162831707f01e79b8da.png.

13.9 The Divergence Theorem

Let calculus_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png be a simple solid region and let calculus_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png be the boundary surface of calculus_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png, given with positive (outward) orientation. Let calculus_1f15588d4acc55769d76ba21967dd6a42b708989.png be a vector field whose component functions have continuous partial derivatives on an open region that contains calculus_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png. Then

calculus_9c4052b7eaf8e02c36ce6ca6bda5006ceab7d1ce.png

Thus the Divergence Theorem states that, under the given conditions, the flux of calculus_1f15588d4acc55769d76ba21967dd6a42b708989.png across the boundary surface of calculus_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is equal to the triple integral of the divergence of calculus_1f15588d4acc55769d76ba21967dd6a42b708989.png over calculus_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.