Evaluate the integral, \int \overrightarrow{ r } \cdot \overrightarrow{ d } r , where C is the helical path described by, x = cost, y = sint, z = t, joining the points given by t = 0 and t = π/2
Question 1.5.2: Evaluate the integral, ∫ r→.d→r , where C is the helical pat...
In Cartesian coordinates,
\begin{aligned}&\vec{r}=x \hat{a}_{x}+y \hat{a}_{y}+z \hat{a}_{z} \\&d \vec{r}=d x \hat{a}_{x}+d y \hat{a}_{y}+d z \hat{a}_{z} \\&I=\int_{C} \bar{r} \cdot d \overline{r} =\int_{C}(x d x+y d x+z d z)=\frac{1}{2}\left[x^{2}+y^{2}+z^{2}\right]_{P}^{Q} \\&\text { with, } t=O P\left(y_{1} y_{1} z\right)=(1,0,0) \\&\text { with, } t=\frac{\pi}{2} Q(x, y, z)=(0,1, \pi / 2) \\&I=\frac{1}{2}\left[x^{2}\right]_{1}^{a}+\frac{1}{2}\left[y^{2}\right]_{0}^{1}+\frac{1}{2}\left[z^{2}\right]_{0}^{\pi / 2}=\frac{\pi^{2}}{8}\end{aligned}Related Answered Questions
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