The path
can be parameterized by the vector-valued function,
![\mathbf r(t)=t(2\,\mathbf i+3\,\mathbf j+9\,\mathbf k)](https://tex.z-dn.net/?f=%5Cmathbf%20r%28t%29%3Dt%282%5C%2C%5Cmathbf%20i%2B3%5C%2C%5Cmathbf%20j%2B9%5C%2C%5Cmathbf%20k%29)
where
. Then
![\mathrm d\mathbf r=(2\,\mathbf i+3\,\mathbf j+9\,\mathbf k)\,\mathrm dt](https://tex.z-dn.net/?f=%5Cmathrm%20d%5Cmathbf%20r%3D%282%5C%2C%5Cmathbf%20i%2B3%5C%2C%5Cmathbf%20j%2B9%5C%2C%5Cmathbf%20k%29%5C%2C%5Cmathrm%20dt)
The line integral reduces to
![\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=\int_{t=0}^{t=1}(x(t)^3\,\mathbf i+y(t)^2\,\mathbf j+z(t)\,\mathbf k)\cdot(2\,\mathbf i+3\,\mathbf j+9\,\mathbf k)\,\mathrm dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_%7B%5Cmathcal%20C%7D%5Cmathbf%20f%5Ccdot%5Cmathrm%20d%5Cmathbf%20r%3D%5Cint_%7Bt%3D0%7D%5E%7Bt%3D1%7D%28x%28t%29%5E3%5C%2C%5Cmathbf%20i%2By%28t%29%5E2%5C%2C%5Cmathbf%20j%2Bz%28t%29%5C%2C%5Cmathbf%20k%29%5Ccdot%282%5C%2C%5Cmathbf%20i%2B3%5C%2C%5Cmathbf%20j%2B9%5C%2C%5Cmathbf%20k%29%5C%2C%5Cmathrm%20dt)
![=\displaystyle\int_{t=0}^{t=1}((2t)^3\,\mathbf i+(3t)^2\,\mathbf j+9t\,\mathbf k)\cdot(2\,\mathbf i+3\,\mathbf j+9\,\mathbf k)\,\mathrm dt](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cint_%7Bt%3D0%7D%5E%7Bt%3D1%7D%28%282t%29%5E3%5C%2C%5Cmathbf%20i%2B%283t%29%5E2%5C%2C%5Cmathbf%20j%2B9t%5C%2C%5Cmathbf%20k%29%5Ccdot%282%5C%2C%5Cmathbf%20i%2B3%5C%2C%5Cmathbf%20j%2B9%5C%2C%5Cmathbf%20k%29%5C%2C%5Cmathrm%20dt)
![=\displaystyle\int_{t=0}^{t=1}(16t^3+27t^2+81t)\,\mathrm dt=\frac{107}2](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cint_%7Bt%3D0%7D%5E%7Bt%3D1%7D%2816t%5E3%2B27t%5E2%2B81t%29%5C%2C%5Cmathrm%20dt%3D%5Cfrac%7B107%7D2)
Answer:
15 minutes
Step-by-step explanation:
Their rate together is 1/6 floor per minute.
Her rate alone is 1/10 floor per minute.
His rate is the difference of these:
1/6 - 1/10 = 5/30 -3/30 = 2/30 = 1/15 . . . . floor per minute
It would take Jacob 15 minutes to do it alone.
Answer:
b i am pretty sure
Step-by-step explanation:
Answer:
{y: y = –7, –2, 3, 8}
Step-by-step explanation:
Subshells are designated using orbital quantum numbers.
<h3>What are quantum numbers?</h3>
Quantum numbers are values that describe an electron's position within the atom. There are four sets of quantum numbers which are;
- Principal quantum number
- Magnetic quantum number
- Spin quantum number
- Orbital quantum number
Subshells are designated using orbital quantum numbers. The question is incomplete hence the quantum numbers were not shown.
Learn more about quantum numbers: brainly.com/question/18835321