1. Let f and g be continuously differentiable functions. PROVE: div( f vg - g v f) = fag -gAf Hint - Use Thm 4: If ü is a vector field an f is a scalar field, then div(f ü)% = f div ü + Vf ü %3D

Q: How do you graph the vector field F = ⟨ƒ(x, y), g(x, y)⟩?

A: Given: vector field

Q: poin) Determine whether the line integral of each oriented vector field (in blue) along the…

A: 1) line integral for vector field Is positive So answer is positive  2) line integral for vector…

Q: Sketch the vector fields. Use a table for it.    F(x,y)=<x,y-x

A: Sketch the vector field Make a table to sketch the vector field. Table should have values (x,y)

Q: (c) If and are smooth scalar fields, show that V x (6V) = Vox V

A:

Q: vector field F = ⟨b cos(ab) + 2a + b, a cos(ab) + 2b + a⟩. State if it is conservative with reasons.

A: The given vector field If the vector field  is conservative Then

Q: 4. Let ƒ,g be C² scalar fields on R³. Verify that V x (ƒVg) = Vƒ × Vg. V × (ƒVg+gVf). V· (SVg) =…

A: As give by the question There are given that the scalar field is f, g be C2 on ℝ3. And, In the above…

Q: Let F and G be vector fields with differentiable components. Express curl (F × G) in term of div and…

A: Let F and G be vector fields with differentiable components.Let We will compute .The cross product…

Q: 77% 77%

A: Given: F(x,y) = 6i-2j

Q: For all differentiable vector fields v(r) and w(r), use index notation to show that - . - . v × (▼ ×…

A: We have to show .

Q: For any vector field F, show that div(curl F) = 0.

A: We need to prove that for any vector field F,  div(curl F) = 0.

Q: Sketch the vector field. F(z, y) = (- 2 2 2 0 2 -2 0 -2 2 -2 2 -2 -2 2 -2 4- -4 -3 -2 -1 2 3 -2 -3-…

A:

Q: Which of the following expressions are meaningful (where F is a vector field and f is a function)?…

A: It is known that, divergence is the net flow of a vector/fluid from a particular area, while Curl…

Q: Show that Vx (V.f) = 0 for any vector field f.

A:

Q: Show that V² (v-x) = 2V.v+x-V²v, where v is a vector field and x is the position vector of a point…

A:

Q: The figure shows a vector field F and three paths from P (-3,0) to Q = (3,0). The top and bottom…

A: Since you have posted multiple questions, we will solve only first three subparts of question for…

Q: grad (фu) —D и @ grad ф + ф grad u

A:

Q: Let be a vector field and fbe a scalar point function, then curl (f) is equivalent to (1) 2. (grad…

A:

Q: Sketch the vector field in the xy-plane. 7(7) = -7/11711³ O O y y X X O O y y X X

A: F→r→=-r→r→3

Q: Find the curl of the vector field F = %3D curl F = %3D

A:

Q: Let (z, y, z)= + z ln (y + 2) be a scalar field. Find the directional derivative of at P(2,2,-1) in…

A: The objective of this question is to find the directional derivative of the scalar field z * ln(y +…