Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Calculus Volume 3

Key Equations

Calculus Volume 3Key Equations

Key Equations

Vector field in 22 F(x,y)=P(x,y),Q(x,y)F(x,y)=P(x,y),Q(x,y)
Vector field in 33 F(x,y,z)=P(x,y,z),Q(x,y,z),R(x,y,z)F(x,y,z)=P(x,y,z),Q(x,y,z),R(x,y,z)
Calculating a scalar line integral Cf(x,y,z)ds=abf(r(t))(x(t))2+(y(t))2+(z(t))2dtCf(x,y,z)ds=abf(r(t))(x(t))2+(y(t))2+(z(t))2dt
Calculating a vector line integral CF·ds=CF· Tds=abF (r(t))· r(t)dtCF·ds=CF· Tds=abF (r(t))· r(t)dt
Calculating flux CF·n(t)n(t)ds=abF(r(t))·n(t)dtCF·n(t)n(t)ds=abF(r(t))·n(t)dt
Fundamental Theorem for Line Integrals Cf·dr=f(r(b))f(r(a))Cf·dr=f(r(b))f(r(a))
Circulation of a conservative field over curve C that encloses a simply connected region Cf·dr=0Cf·dr=0
Green’s theorem, circulation form CPdx+Qdy=DQxPydA,CPdx+Qdy=DQxPydA, where C is the boundary of D
Green’s theorem, flux form CF·dr=DQxPydA,CF·dr=DQxPydA, where C is the boundary of D
Green’s theorem, extended version DF·dr=DQxPydADF·dr=DQxPydA
Curl ×F=(RyQz)i+(PzRx)j+(QxPy)k×F=(RyQz)i+(PzRx)j+(QxPy)k
Divergence ·F=Px+Qy+Rz·F=Px+Qy+Rz
Divergence of curl is zero ·(×F)=0·(×F)=0
Curl of a gradient is the zero vector ×(f)=0×(f)=0
Scalar surface integral Sf(x,y,z)dS=Df(r(u,v))||tu×tv||dASf(x,y,z)dS=Df(r(u,v))||tu×tv||dA
Flux integral SF·NdS=SF·dS=DF(r(u,v))·(tu×tv)dASF·NdS=SF·dS=DF(r(u,v))·(tu×tv)dA
Stokes’ theorem CF·dr=ScurlF·dSCF·dr=ScurlF·dS
Divergence theorem EdivFdV=SF·dSEdivFdV=SF·dS
Order a print copy

As an Amazon Associate we earn from qualifying purchases.


This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at
Citation information

© Feb 5, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.