A scalar potential φ has the following gradient:vφ

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A scalar potential φ has the following gradient:
vφ = yZî + xZĵ + xyk̂ . Consider the integral
∫_c φ.dr on the curve r = xî + yĵ + ẑ
The curve C is parameterized as follows:
x = t and 1 ≤ t ≤ 3
y = t
z = 3t²

The value of the integral is ________

1. 721
2. 722
3. 716
4. 726

Correct Option: D

∫_c ∇Φ . dr = ∫_c(yzî + xzĵ + xyk̂)(dxî + dyĵ + dzk̂
= ∫_c (yzdx + xzdy + xydz) = xyz
Given that x = t, y = t², z = 3t²
= t.t² . 3t²|₁³
= 3(t⁵) |₁²
= 3(3⁵ - 1)
= 726

		x = t	and 1 ≤ t ≤ 3
		y = t
		z = 3t²