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Calculus Volume 3

Key Equations

Calculus Volume 3Key Equations

Key Equations

Vector-valued function r(t)=f(t)i+g(t)jorr(t)=f(t)i+g(t)j+h(t)k,orr(t)=f(t),g(t)orr(t)=f(t),g(t),h(t)r(t)=f(t)i+g(t)jorr(t)=f(t)i+g(t)j+h(t)k,orr(t)=f(t),g(t)orr(t)=f(t),g(t),h(t)
Limit of a vector-valued function limtar(t)=[limtaf(t)]i+[limtag(t)]jorlimtar(t)=[limtaf(t)]i+[limtag(t)]j+[limtah(t)]klimtar(t)=[limtaf(t)]i+[limtag(t)]jorlimtar(t)=[limtaf(t)]i+[limtag(t)]j+[limtah(t)]k
Derivative of a vector-valued function r(t)=limΔt0r(t+Δt)r(t)Δtr(t)=limΔt0r(t+Δt)r(t)Δt
Principal unit tangent vector T(t)=r(t)r(t)T(t)=r(t)r(t)
Indefinite integral of a vector-valued function [f(t)i+g(t)j+h(t)k]dt=[f(t)dt]i+[g(t)dt]j+[h(t)dt]k[f(t)i+g(t)j+h(t)k]dt=[f(t)dt]i+[g(t)dt]j+[h(t)dt]k
Definite integral of a vector-valued function ab[f(t)i+g(t)j+h(t)k]dt=[abf(t)dt]i+[abg(t)dt]j+[abh(t)dt]kab[f(t)i+g(t)j+h(t)k]dt=[abf(t)dt]i+[abg(t)dt]j+[abh(t)dt]k
Arc length of space curve s=ab[f(t)]2+[g(t)]2+[h(t)]2dt=abr(t)dts=ab[f(t)]2+[g(t)]2+[h(t)]2dt=abr(t)dt
Arc-length function s(t)=at(f(u))2+(g(u))2+(h(u))2duors(t)=atr(u)dus(t)=at(f(u))2+(g(u))2+(h(u))2duors(t)=atr(u)du
Curvature κ=T(t)r(t)orκ=r(t)×r″(t)r(t)3orκ=|y|[1+(y)2]3/2κ=T(t)r(t)orκ=r(t)×r″(t)r(t)3orκ=|y|[1+(y)2]3/2
Principal unit normal vector N(t)=T(t)T(t)N(t)=T(t)T(t)
Binormal vector B(t)=T(t)×N(t)B(t)=T(t)×N(t)
Velocity v(t)=r(t)v(t)=r(t)
Acceleration a(t)=v(t)=r″(t)a(t)=v(t)=r″(t)
Speed v(t)=v(t)=r(t)=dsdtv(t)=v(t)=r(t)=dsdt
Tangential component of acceleration aT=a·T=v·avaT=a·T=v·av
Normal component of acceleration aN=a·N=v×av=a2aT2aN=a·N=v×av=a2aT2
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