A scalar potential φ has the following gradient: ∇ϕ = yz î + xz ĵ + xy k̂. Consider the integral \int _{c}^{}⁡∇ ϕ .d⃗r on the curve r^{\to} = xi^{\^} + yj^{\^} + zk^{\^} The curve C is parameterized as follows: {\\begin{array}{c}x=t \ y=t^2 \ z=3 t^2\end{array} and 1 \leq t \leq 3 The value of the integral is __________.

GATE Mechanical Engineering (ME) 2016 Shift 2 Solved Paper

Question : 36 of 65

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A scalar potential φ has the following gradient: ∇ϕ = yz î + xz ĵ + xy k̂. Consider the integral

∫ _{c}^{}⁡∇ ϕ .d⃗r

on the curve

r↖{→} = xi↖{\^} + yj↖{\^} + zk↖{\^}

The curve C is parameterized as follows:

{\{\table x=t ; y=t^2 ; z=3 t^2}

and

1 ≤ t ≤ 3

The value of the integral is __________.

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