Indefinite integral of cross product of 2 function

  • #1
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Homework Statement:

If ##\vec{v} (t)## is a vector function of t, find indefinite integral
$$\int {\left(\vec v \times \frac{d^2 \vec v}{dt^2} dt\right)}$$

Relevant Equations:

Idk.
I've tried with this work in attachment. i&m not sure of my answer is correct.
 

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  • #2
STEMucator
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The problem statement is not given properly. Can you provide us proper details? If you are computing:

$$\int v \frac{d^2v}{dt^2} dt$$​

You mentioned ##dv = b(t) dt##, and then computed ##v##:

$$v = \frac{dv}{dt}$$​

You know this is inherently not correct, unless the function is specifically ##v(t) = e^t##. Don't forget about generality when writing math proofs, or computing such things.
 
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  • #3
haruspex
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The problem statement is not given properly. Can you provide us proper details? If you are computing:
$$\int v \frac{d^2v}{dt^2} dt$$​
The thread title says cross product.
 
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  • #4
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Screen Shot 2020-06-21 at 10.51.10 AM.png
Screen Shot 2020-06-21 at 10.51.39 AM.png

Integrate all the remaining terms, and you get a sum of antiderivatives that subtract from the other terms.

I still dislike how we use X for multiplication, and cross product every time I see it, sorry for the confusion.
 
  • #5
haruspex
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@agnimusayoti , I think you confused everyone with your unfortunate choice of notation, using v (no arrow) for ##\frac{d\vec v}{dt}## and u for ##\vec v##.
Where your algebra goes wrong is in handling ##\int \frac{d\vec v}{dt}\times d\vec v##. The correct result becomes obvious if you write it as ##\int \frac{d\vec v}{dt}\times \frac{d\vec v}{dt}dt##
 

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