Question 27.6: A curve, C, is defined parametrically by x = 4 y = t³ z =......
A curve, C, is defined parametrically by
x = 4 y = t³ z = 5 + t
and is located within a vector field F = yi + x² j + (z + x)k.
(a) Find the coordinates of the point P on the curve where the parameter t takes the value 1.
(b) Find the coordinates of the point Q where the parameter t takes the value 3.
(c) By expressing the line integral \int_C \mathbf{F} \cdot \mathrm{d} \mathbf{s} entirely in terms of t find the value of the line integral from P to Q. Note that ds = dxi + dyj + dzk.
(a) When t = 1, x = A, y = 1 and z = 6, and so P has coordinates (4, 1, 6).
(b) When r = 3, x = 4, y = 27 and z = 8, and so Q has coordinates (4, 27, 8).
(c) To express the line integral entirely in terms of t we note that if x = 4, \frac{\mathrm{d} x}{\mathrm{~d} t}=0 \text { so that } \mathrm{d} x \text { is also zero. If } y=t^3 \text { then } \frac{\mathrm{d} y}{\mathrm{~d} t}=3 t^2 \text { so that } \mathrm{d} y=3 t^2 \mathrm{~d} t \text {. }
Similarly since z=5+t, \frac{\mathrm{d} z}{\mathrm{~d} t}=1 \text { so that } \mathrm{d} z=\mathrm{d} t. The line integral becomes