Question

1. A trail of ants on a silo is described by the helix ?: ?⃗(?) = (cos ?)?̂+ (sin ?)?̂+ (3?)?̂ , for 0 ≤ ? ≤ 4?. The “linear ant density” along the trail is given by : ?(?, ?, ?) = 5? 2 + 5? 2 + 12? ???? ? . Evaluate the line integral : ∫ ?(?, ?, ?) ? ?? , and describe what this value represents.

2. Given the function: ?(?, ?) = 2?? − ?² − ?²

(a) Give the formula for the gradient vector field ?⃗(?, ?) = ∇?(?, ?)

(b) Describe and sketch the vector field along both coordinate axes and along the lines ? = ±?.

(c) Compute the divergence and the curl of the vector field. (d) Evaluate the line integral of the vector field along any curve C from the point ?(−2,0) to the point ?(1, 5) using the Fundamental Theorem of Line Integrals.

Answer 1