Calculus Volume 3

# Key Equations

Calculus Volume 3Key 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
$limt→ar(t)=[limt→af(t)]i+[limt→ag(t)]jorlimt→ar(t)=[limt→af(t)]i+[limt→ag(t)]j+[limt→ah(t)]klimt→ar(t)=[limt→af(t)]i+[limt→ag(t)]jorlimt→ar(t)=[limt→af(t)]i+[limt→ag(t)]j+[limt→ah(t)]k$
• Derivative of a vector-valued function
$r′(t)=limΔt→0r(t+Δt)−r(t)Δtr′(t)=limΔt→0r(t+Δt)−r(t)Δt$
• 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
$∫ab[f(t)i+g(t)j+h(t)k]dt=[∫abf(t)dt]i+[∫abg(t)dt]j+[∫abh(t)dt]k∫ab[f(t)i+g(t)j+h(t)k]dt=[∫abf(t)dt]i+[∫abg(t)dt]j+[∫abh(t)dt]k$
• Arc length of space curve
$s=∫ab[f′(t)]2+[g′(t)]2+[h′(t)]2dt=∫ab‖r′(t)‖dts=∫ab[f′(t)]2+[g′(t)]2+[h′(t)]2dt=∫ab‖r′(t)‖dt$
• Arc-length function
$s(t)=∫at(f′(u))2+(g′(u))2+(h′(u))2duors(t)=∫at‖r′(u)‖dus(t)=∫at(f′(u))2+(g′(u))2+(h′(u))2duors(t)=∫at‖r′(u)‖du$
• 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
$aT=a·T=v·a‖v‖aT=a·T=v·a‖v‖$
• Normal component of acceleration
$aN=a·N=‖v×a‖‖v‖=‖a‖2−aT2aN=a·N=‖v×a‖‖v‖=‖a‖2−aT2$

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