Calculus II - Vector functions

15 flashcards · Mathematics · by luckydog789 (harrison holbrook)

1. Vector Function

   A vector-valued function of a real variable, or simply a vector function, r=r(t)r=r(t)r=r(t) , is a function whose domain is a subset of the set of real numbers and whose range is a set of vectors (one-dimensional matrices). These vector functions are often represented as r=r(t)=⟨x(t),y(t)⟩r=r(t)=\left\langle x(t),y(t) \right\rangler=r(t)=⟨x(t),y(t)⟩ , where x=x(t)x=x(t)x=x(t) and y=y(t)y=y(t)y=y(t) are real functions defined over the interval.

2. Limit of a vector function

   For a vector function r(t)=⟨x(t),y(t)⟩r(t)=\left\langle x(t),y(t) \right\rangler(t)=⟨x(t),y(t)⟩ , lim⁡t→t0r(t)=⟨lim⁡t→t0x(t),lim⁡t→t0y(t)⟩\lim_{t \to t_0} r(t)=\left\langle \lim_{t \to t_0}x(t),\lim_{t \to t_0}y(t) \right\ranglelimt→t0​​r(t)=⟨limt→t0​​x(t),limt→t0​​y(t)⟩

3. Continuity of a vector function at a number

   A vector function r=r(t)r=r(t)r=r(t) is continuous at a real number t0t_0t0​ if r(t0)r(t_0)r(t0​) is defined lim⁡t→t0r(t)\lim_{t \to t_0} r(t)limt→t0​​r(t) exists lim⁡t→t0r(t)=r(t0)\lim_{t \to t_0} r(t)=r(t_0)limt→t0​​r(t)=r(t0​)

4. Derivative of a vector function

   The derivative of a vector function r=r(t)r=r(t)r=r(t) is r′(t)=drdt=lim⁡h→0r(t+h)−r(t)h=⟨x′(t),y′(t)⟩r'(t)=\frac{dr}{dt}=\lim_{h\to 0}\frac{r(t+h)-r(t)}{h}=\left\langle x'(t),y'(t) \right\rangler′(t)=dtdr​=limh→0​hr(t+h)−r(t)​=⟨x′(t),y′(t)⟩

5. Sum rule of vector function derivatives

   ddx[u(t)+v(t)]=ddxu(t)+ddxv(t)\frac{d}{dx}[u(t)+v(t)]=\frac{d}{dx}u(t)+\frac{d}{dx}v(t)dxd​[u(t)+v(t)]=dxd​u(t)+dxd​v(t)

6. Scalar multiplication rule of vector function derivatives

   ddx[f(t)u(t)]=f′(t)u(t)+f(t)u′(t)\frac{d}{dx}[f(t)u(t)]=f'(t)u(t)+f(t)u'(t)dxd​[f(t)u(t)]=f′(t)u(t)+f(t)u′(t)

7. Chain rule of vector function derivatives

   ddx[u(f(t))]=u′(f(t))f′(t)\frac{d}{dx}[u(f(t))]=u'(f(t))f'(t)dxd​[u(f(t))]=u′(f(t))f′(t)

8. Tangent vector to a curve

   Suppose r=r(t)r=r(t)r=r(t) is a vector function defined on the interval [a,b][a,b][a,b] and differentiable on the interval (a,b)(a,b)(a,b) , Let P0P_0P0​ be the point on the curve CCC traced out by r=r(t)r=r(t)r=r(t) corresponding to t=t0t=t_0t=t0​ , a<t0<ba<t_0<ba<t0​<b . If r′(t0)≠0r'(t_0) \neq 0r′(t0​)=0 , then the vector r′(t0)r'(t_0)r′(t0​) is a tangent vector to the curve at t0t_0t0​ . The line containing P0P_0P0​ in the direction of r′(t0)r'(t_0)r′(t0​) is the tangent line to the curve…

9. Arc length of a vector function

   If a smooth curve CCC is traced out by the vector function r=r(t)r=r(t)r=r(t) over a≤t≤ba \leq t \leq ba≤t≤b , then the arc length sss along CCC from t=at=at=a to t=bt=bt=b is s=∫ab∣∣r′(t)∣∣dt=∫ab(dxdt)2+(dydt)2dts=\int_a^b||r'(t)||dt=\int_a^b\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}dts=∫ab​∣∣r′(t)∣∣dt=∫ab​(dtdx​)2+(dtdy​)2 ​dt

10. Norm of vector

    ∣∣⟨x(t),y(t)⟩∣∣=(dxdt)2+(dydt)2||\left\langle x(t),y(t) \right\rangle||=\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}∣∣⟨x(t),y(t)⟩∣∣=(dtdx​)2+(dtdy​)2 ​

11. Velocity vector

    First derivative of the position vector

12. Acceleration vector

    Second derivative of the position vector

13. Speed

    Norm of the velocity vector

14. Integral of vector function

    ∫r(t)dt=⟨∫x(t)dt,∫y(t)dt⟩\int r(t)dt=\left\langle \int x(t) dt, \int y(t) dt \right\rangle∫r(t)dt=⟨∫x(t)dt,∫y(t)dt⟩

15. Always remember to account for the ... when solving vector differential equations or definite integrals

    Initial conditions