Which of the following functions V can be considered a vector field. Select all that apply. V: R² V: R² V: R³ R² R³ → R³ V: R³ → R¹ R³ OV: R¹

Consider the vector field shown in the following arrow map. Y ↑ ↑ V 5 3 1 x 2 3 4 5 Select the option that gives the vector field that could correspond to this arrow map. Select one: (y-4)i + (2-x) j (x-4)i + (2-y) j O (4-x)i + (y-2) j O (4-y)i + (x - 2) j

Does it make sense? Are the following expressions defined? If so, state whether the result is a scalar or a vector. Assume F is a sufficiently differentiable vector field and w is a sufficiently differentiable scalar-valued function.a. ∇ ⋅ φ          b. ∇F                   c. ∇ ⋅ ∇φd. ∇(∇ ⋅ φ)     e. ∇(∇ x φ)          f. ∇ ⋅ (∇ ⋅ F)g. ∇ x ∇φ       h. ∇ x (∇ ⋅ F)        i. ∇ x (∇ x F)

Show that if F = xi+yj + zk, then VXF=0. The curl of a vector field F = Mi+Nj + Pk is as follows. ap ƏN ?M ap dy dz dz ?x Use the formula above to find the curl of F=xi+yj + zk. curl F = VXF= VxF = V x (xi+yj+zk) dz dy = - i+ dy dz = i+ İ+ dx dz dz ?x + k j+ dy ax ?x ду ƏN ax k ƏM ду