vector calculus

(a)

Parameterise the path

$\mathbf{x}(t)=(t,t,t) \quad 0 \le t \le 1$

In terms of the parameter

$\mathbf{F}(\mathbf{x}) = \begin{pmatrix} 2 \alpha t + \alpha^2 t + \alpha t \\ \alpha^2 t + \beta t \\ \alpha t + \beta t^2 \end{pmatrix}$

Also

$\frac{d}{dt}\mathbf{x}(t) = (1,1,1)$

$\begin{aligned} \int_\Gamma \mathbf{F} \cdot d\mathbf{x} &= \int_0^1 \begin{pmatrix} 2 \alpha t + \alpha^2 t + \alpha t \\ \alpha^2 t + \beta t \\ \alpha t + \beta t^2 \end{pmatrix} \cdot (1,1,1) \, dt \\ &= \int_0^1 (4\alpha t + 2 \alpha^2 t + \beta t + \beta t^2) \, dt \\ &= \left[ 2\alpha t^2 + \alpha^2 t^2 + \beta t^2/2 + \beta t^3/3 \right]_0^1 \\ &= 2\alpha + \alpha^2 + \beta /2 + \beta /3 \end{aligned}$

Thus

$\int_\Gamma \mathbf{F} \cdot d\mathbf{x} = 2\alpha + \alpha^2 + \frac{5}{6}\beta$

NST First Year Maths

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2019 Paper 2 Question 13