Answer: a) Fdr = 0 b) Fdr = 0
Step-by-step explanation:
According to the integral Fdr
a) If r=t²i + t²j
If the field F(x, y, z) = y sin(z)i + x sin(z)j + xy cos(x)k
Integral Fdr = y sin(z)i + x sin(z)j + xy cos(x)k d(t²i + t²j)
Multiplying component wise.
Fdr = y sin(z)i(dt²i) + x sin(z)j(dt²j) + xy cos(x)k d(0k)
Remember that dot product of different components gives 'zero' i.e i.j = 0 etc.
Fdr = ysin(z)dt² + xsin(z)dt² + 0...(1)
Generally for 2dimensional object r=xi+yj
Comparing this to r = t²i+t²j
x=t² and y = t² z = 0...(2)
Substituting 2 into 1
Fdr = t²sin0dt² + t²sin0dt²
Integrating resulting Fdr with respect to t² within the limit 0 ≤ t ≤ 2, we have
Fdr = 0
b) since there is no z-component in the second part as well, integral of Fdr within the limit 0 ≤ t ≤ 1 will also be 0.
