Vector Analysis-Gradient Divergence and Curl, guaranteed and verified 100% Pass

Gradient, Divergence, and Curl



Def. A Vector Field in ℝ𝑛 , is a map 𝐹⃗ : 𝐴 ⊂ ℝ𝑛 → ℝ𝑛 , that assigns to each
point 𝑥 = (𝑥1 , 𝑥2 , 𝑥3 , … , 𝑥𝑛 )𝜖𝐴, a vector 𝐹⃗ (𝑥 )𝜖ℝ𝑛 .

If 𝑛 = 2 we call 𝐹⃗ a vector field in the plane.

If 𝑛 = 3 we call 𝐹⃗ a vector field in space.



We can always write a vector field in space in the form:

𝐹⃗ (𝑥, 𝑦, 𝑧) = 𝐹1 (𝑥, 𝑦, 𝑧)𝑖⃗ + 𝐹2 (𝑥, 𝑦, 𝑧)𝑗⃗ + 𝐹3 (𝑥, 𝑦, 𝑧)𝑘⃗⃗, or

𝐹⃗ (𝑥, 𝑦, 𝑧) =< 𝐹1 (𝑥, 𝑦, 𝑧), 𝐹2 (𝑥, 𝑦, 𝑧), 𝐹3 (𝑥, 𝑦, 𝑧) >

Notice that this is different from a real-valued function from ℝ3 → ℝ

(which we will sometimes call a Scalar Field).


Ex. 𝐹⃗ (𝑥, 𝑦) = 2𝑖⃗ − ⃗⃗⃗⃗
3𝑗 = < 2, −3 > is a vector field in the plane drawn as

, 2


Ex. 𝐹⃗ (𝑥, 𝑦) = 𝑦𝑖⃗ − 𝑥𝑗⃗ = < 𝑦, − 𝑥 > is a vector field in the plane

Point 𝐹⃗ (𝑥, 𝑦)

(1,0) < 0, −1 >

(1, −1) < −1, −1 >

(0, −2) < −2,0 >

(−2, −2) < −2,2 >

(−4,0) < 0,4 >

(−4,4) < 4,4 >

(0,8) < 8,0 >

(8,8) < 8, −8 >




⃗⃗ is a vector field on ℝ3 .
Ex. 𝐹⃗ (𝑥, 𝑦, 𝑧) = (𝑥 2 𝑧)𝑖⃗ + 𝑒 𝑦 𝑗⃗ + sin(𝑥𝑧)𝑘

𝑓 (𝑥, 𝑦, 𝑧) = 𝑥 2 𝑧 + 𝑒 𝑦 + sin(𝑥𝑧) is a real-valued function on ℝ3 .

Notice that for every value of 𝑥, 𝑦, 𝑧 , 𝐹⃗ (𝑥, 𝑦, 𝑧) gives us a vector in ℝ3 .

For every value of 𝑥, 𝑦, 𝑧, 𝑓 (𝑥, 𝑦, 𝑧) gives us a real number, not a vector in ℝ3 .