f :V → R given by T f(x) = v"x, where V C R" is any subspace, and v E V is any fixed vector.

Q: Find the gradient fields of the functions g(x, y, z) = xy + yz + xz

A:

Q: Match the vector field with its graph. F(x, y) = xi + 3yj + +x ' ||||||||||| 畫 至 X

A: Solution is given below:

Q: Find the gradient vector field of f. f(x, y, z) = 2√x² + y² + z²

A:

Q: Find the gradient fields of g(x, y, z) = xy + yz + xz

A: Given that, g(x)=xy+yz+xz   We have to find, Gradient Field.   As we know,  grad…

Q: Find the domain of r(t) and the interval(s) on which the vector- valued function is continuous. r…

A: We need to find the domain and the interval on which the vector-valued function is continuous.

Q: Determine the interval (s) on which the vector-valued function r(t) = ti + Vt + 1j + (t2 + 1)k is…

A:

Q: Evaluate the integral curves for the vector field F(x,y) : i – 4xy²j . О a. - 4у — х2 = C 4x2y2 Ob.…

A:

Q: Consider a conservative vector field F(x, y) = (y + 6xy^3 )i + (x + y + 9x^2 y^2 )j (a) Find a…

A: Given: A conservation vector field F(x,y)=y+6xy3i^+x+y+9x2y2j^. To findf=∇f.

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = ln(x + 5y + 6z). F(x, y, z) = =

A:

Q: Q. No. 1): Let H ={ h : x* + y s 0.9 + 175 Show that "H" is not subspace of R².

A: Let , V be a vector space. A subset S of V is said to be a subspace of V if : 1) for any a, b ∈S,…

Q: Integrate the vector field h (x, y, =) = xy i +yz j+ xz k over r (1) =ui+ử j+ư k, u E[-1,1]. a) 0 b)…

A: According to the given information, it is required to integrate the vector field over the vector…

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = tan(6r + 2y + 2) . F(2, y, 2) =

A: Find the gradient vector field

Q: If a vector function v(t) = (t, Int,t-²) is given for a scalar function Ø = x In z + Jy - cos(x + 1)…

A: Given the scalar function φ=xlnz+y-cosx+1. This scalar field is to be evaluated over the path…

Q: Compute the gradient vector fields of the following functions: A. f(x, y) = 1x² +10y² Vf(x,y) =i+j…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub parts for…

Q: Let U ⊂ Rn be open and f : U → Rm be differentiable at a ∈ U. Prove that for any vector ~v ∈ Rn,…

A: To Determine :-  Let be open and be differentiable at . Prove that for any vector .

Q: Find the gradient vector field of f(x, y, z) = x³y* + y° z³ + xz°.

A:

Q: Find the gradient vector field of the function f. f(x, y, z) = 6ze-xy Vf(x, y, z) =

A: To find the gradient vector field of the given function f.

Q: In the vector field F(x, y) = i + j Compute ||F||. Sketch several representative vectors in the…

A: Given : F(x, y) = i + j

Q: Find the gradient vector field Vf of f. 1 f(r, y) = (y – 2)?

A: For a given vector fx, y the gradient vector denoted by ∇f is ∂ ∂ xf∂ ∂ yfSo,=∂ ∂ x12y-x2∂ ∂…

Q: Find the gradient vector field of f(x, y, z) = e3r + 8y+6z

A:

Q: Show that the vector field F (x, y, z) = xz i + xyz j y 2 k is not conservative.

A: Consider the following vector: Fx,y,z=xzi^+xyzj^+y2k^ Find the Curl of F→:…

Q: 13

A: We have to find out conservative or not

Q: Find the gradient vector field of f(x, y, z) 5x+ 2y+8z

A:

Q: Find the streamlines of the vector field and the streamline through the given point. F(x, y, z) =i+…

A:

Q: Integrate the vector field h (x, y, =)=ry i+yz j+x k over r (1u) = ui+ j+k, -ןue-L1

A:

Q: F=(P(x.y).Q(x. y).0) Find when the Vector field F is conservative.

A:

Q: Find a non-zero and non-constant vector function F(x, y) = F(x, y)i+ F2(x, y)j, %3D OF OF F such…

A: Given, to find a non-zero and non-constant vector function in the form…

Q: Select the irrotational vector field. O A yz î-2xz + xyz k O B. yz î + xz î + xz R OCzî + z2 + yz R…

A:

Q: (c) Determine the flux of the vector field F = x³î – 2yĵ + 7z²k across a cone z = J 8 by using…

A:

Q: The circuitation of the vector field F(x, y) = (x- x'y, xy) along the boundary of {(x, y) ER: x + (y…

A:

Q: Find a non-zero and non-constant vector function F(x, y) = F1(x, y)i+ F2(x, y)j, ƏF OF 02F and dy?…

A:

Q: 2x Find the gradient vector field (F(, y, z)) of f(x, y, z) = y cos F(2,y, 2) =

A:

Q: -dt. dF = (dr · V)F+ at

A:

Q: Match the vector field with its graph. F(x, y) = xi + 3yj 5

A: We can solve this using by draw a graph

Q: Compute the gradient vector fields of the following functions: A. f(x, y) = 1x² +10y² Vf(x, y) =i+ j…

A: “Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: Consider the vector field i = P(x, y) = –3y+ ¤(x² + y² – 1), ý = Q(x, y) = 3x +y(x² + y² – 1) | %3D

A: consider the following

Q: ) Show that the vector field, F3 yz £ + zx ŷ + xy 2 c

A: Given: The vector field is, F3 = yz x^ + zx y^ + xy z^

Q: Find the gradient vector field ∇f of f(x, y) = x2 − 4y.

A:

Q: Determine the vector field of F. F(x, y) = yl – x) 1.

A: Vector field

Q: Consider the set S = yl E R : 4x +6z=7y, x – 13y=0 . Is this a subspace of R ? Explain why or why…

A:

Q: x – y T x + Y. a) A vector in im(T) that is not 0 is

A:

Q: Find the gradient vector field of f. f(x, y) = xe³xy Vf(x, y) =

A:

Q: x [ũi |….. Jūs]. If the equation AX = 0 5 dimensional subspace of R7. nded to form a basis for R5.…

A:

Q: Determine which of these vector fields are conservative. If a vector field is not conservative,…

A:

Q: Find the gradient vector field of f(x, y) = In(2x + 3y) V

A:

Q: Determine the vector field of F. F(x, y) = yl – xj 1.

A: Solution

Q: Find the gradient vector field of f(x, y, z) = e® 5x+2y+8z

A:

Q: 3. Evaluate |F. dr where C is the curve F(t) = (t, t°, t“), 0<t< 1 and the vector field F(r, y, z) =…

A:

Q: Match the vector field with its graph. F(x, y) = xi + 3yj 進 ↑ -6+ <||||||||||| Hex

A:

Q: Evaluate ∫C F . T ds for the vector field F = x2 i - y j along the curve x = y2 from (4, 2) to (1,…

A: Given information: The value of F=x2i+yj, x=y2 and from (4, 2) to (1, -1). Calculation: Consider…