Solutions to Various Problems in Vector Calculus
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Solution Summary
We solve various problems in vector calculus.
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1. We have c(t) = (x(t), y(t), z(t)) where c(0) = (0, -5, 1) and c'(t) = (x'(t), y'(t), z'(t)) = (t, e^t, t^2).
Thus we have the following three differential equations with given initial conditions:
x'(t) = t; x(0) = 0
y'(t) = e^t; y(0) = -5
z'(t) = t^2; z(0) = 1
The solution to the first equation is x(t) = 1/2 t^2 + C_x. Plugging in the initial condition we have
x(0) = C_x = 0, whence x(t) = 1/2 t^2.
The solution to the second equation is y(t) = e^t + C_y. Plugging in the initial condition we have
y(0) = 1 + C_y = -5, whence C_y = -6 and y(t) = e^t - ...
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