1
Vector Fields- HW Problems
⃗ (𝑥, 𝑦) = (cos(𝑦))𝑖 − (sin(𝑥 ))𝑗 is not a
1. Show that the vector field 𝑉
gradient vector field.
⃗ (𝑥, 𝑦, 𝑧).
2. Find a function 𝑓(𝑥, 𝑦, 𝑧) such that ∇𝑓 (𝑥, 𝑦, 𝑧) = 𝑉
⃗ (𝑥, 𝑦, 𝑧) =< 𝑥 2 , 𝑦 2 , 𝑧 2 >
a. 𝑉
⃗ (𝑥, 𝑦, 𝑧) =< 𝑦𝑐𝑜𝑠(𝑧), 𝑥𝑐𝑜𝑠(𝑧), −𝑥𝑦𝑠𝑖𝑛(𝑧) >
b. 𝑉
3. Show that the given curve 𝑐 (𝑡) is a flow line for the given vector
⃗ (𝑥, 𝑦, 𝑧).
field 𝑉
a. 𝑐 (𝑡) =< 𝑒 𝑡 , cos(𝑡) , sin(𝑡) >; ⃗ (𝑥, 𝑦, 𝑧) =< 𝑥, −𝑧, 𝑦 >
𝑉
1 1
b. 𝑐 (𝑡) =< 2, , 𝑒 2𝑡 >; ⃗ (𝑥, 𝑦, 𝑧) =< −2𝑦 3 , −𝑥, 2𝑧 >.
𝑉
𝑡 𝑡
Vector Fields- HW Problems
⃗ (𝑥, 𝑦) = (cos(𝑦))𝑖 − (sin(𝑥 ))𝑗 is not a
1. Show that the vector field 𝑉
gradient vector field.
⃗ (𝑥, 𝑦, 𝑧).
2. Find a function 𝑓(𝑥, 𝑦, 𝑧) such that ∇𝑓 (𝑥, 𝑦, 𝑧) = 𝑉
⃗ (𝑥, 𝑦, 𝑧) =< 𝑥 2 , 𝑦 2 , 𝑧 2 >
a. 𝑉
⃗ (𝑥, 𝑦, 𝑧) =< 𝑦𝑐𝑜𝑠(𝑧), 𝑥𝑐𝑜𝑠(𝑧), −𝑥𝑦𝑠𝑖𝑛(𝑧) >
b. 𝑉
3. Show that the given curve 𝑐 (𝑡) is a flow line for the given vector
⃗ (𝑥, 𝑦, 𝑧).
field 𝑉
a. 𝑐 (𝑡) =< 𝑒 𝑡 , cos(𝑡) , sin(𝑡) >; ⃗ (𝑥, 𝑦, 𝑧) =< 𝑥, −𝑧, 𝑦 >
𝑉
1 1
b. 𝑐 (𝑡) =< 2, , 𝑒 2𝑡 >; ⃗ (𝑥, 𝑦, 𝑧) =< −2𝑦 3 , −𝑥, 2𝑧 >.
𝑉
𝑡 𝑡
