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

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 13 minutes ago

asked 22 minutes ago

asked 26 minutes ago

asked 30 minutes ago

asked 30 minutes ago

asked 39 minutes ago

asked 39 minutes ago

asked 49 minutes ago

asked 49 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago