is Cantesion Tensor of orden 2 on the form 3 I= 2 a=(3,4, 1) Find - 2 T.a and a. T

Q: Compute the circulation of the vector field F = in moving a particle around the path showed in the…

A: From Green's theorem of circulation form Work done on a particle by force field F→=Px,y,Qx,y is…

Q: A hot-air balloon has the truncated spherical shape. The hot gases escape through the porous…

A: Image is attached with detailed solution.

Q: Let 3 = (X'X)-¹X'Y as shown in class. Let y = w'ß, where w is a column vector. Show that Var(y) =…

A:

Q: Vector fields V and W are defined by V- (2х — Зу + z, -3х — у + 4z,4y + z) W- (2x - 4y — 5z, - 4x +…

A:

Q: A vector field Fis conservative when F is the gradient vector of some potential (scalar) functi O…

A: To see 2nd step

Q: If (a, b) is a local maximum of f, then D, f(a, b) = 0 for any unit vector u e R?. True False

A: Question: if (a,b) is a local maximum of f, then Du→f(a,b)=0 for any unit vector u→∈R2.

Q: Q4) Let V be a vector space over F. Let a E F and v E V satisfy the equation av = 0. Prove that…

A:

Q: 3) Use the basic definition of a Cartesian tensor to show the following: (a) That for any general,…

A: Solution : Consider a rotation of the (unprimed) coordinate axes through an angle θ to give the new…

Q: Let o be the cube with side length 1, faces paralet to the coordinate planes, and centered at the…

A: For the detailed solution of the problem follow the next steps.

Q: Determine the vector field of FF. F(x,y)=8i−4j

A: Solution

Q: Determine it this new def ot O wald make Va Vector Space

A:

Q: G. It Symmetric tansor anti- Symmetric tensor, that Apg is a is an then prove

A:

Q: If Apq is a symmetric tensor, Bpq is an anti-symmetric tensor, then prove that ApqBpq = 0

A: Given two tensors Apq & Bpq such that Apq is symmetric and Bpq is anti-symmetric. To prove that…

Q: If A(r, s, t) be a function of the coordinates in n-dimensional space such that for an arbitrary…

A: Given: Ar,s,t is a function of the coordinates in n-dimensional space such that for an arbitrary…

Q: Suppose that f is a vector of length 8 with F1{feven} = -2 – i and F{f»dd} = 3+i. %3D Compute the…

A:

Q: Consider the following. (a) Show that the set of vectors in R" is orthogonal. (v5, V5. V5) · (0, Vz.…

A: A set of vectors is said to be orthogonal if dot product of any two different vectors of the set is…

Q: yi Determine whether the vector field F(x, y) %3D 1.

A:

Q: v.(OF = (vo) - F+o(V .F (V) F+

A:

Q: Mathematical Definition What is A set S that spans a vector space V?

A: The set S = { v1, v2, v3, .... , vk} is a set of vectors in a vector space V.

Q: Compute the curl of the vector field F = (r', y³, z*). curl(F(r, y, z)) = %3D What is the curl at…

A:

Q: Q4. Consider a vector space that consists of the linear combinations of the following functions: {1,…

A: We will solve this by the help of definition of basis and dimension of vector spaces

Q: Which vector field would generate this graph? 0( - y, – a) O (y, z) O (x, – y) 0( - 1, y)

A: We check the given vector field one by one. First one is fx,y=-y,-x Now, for point 1,1 we see that…

Q: Let U = span(1, 1, 0), (1, 1, 1)). Find a vector u E U such that Ilu – (1, 0, 0)|| is as small as…

A: First, we need to find an orthogonal basis for U. Let us take u1=(1,1,0) and v2=(1,1,1) Now, let us…

Q: Determine the vector field of F. F(x, y) = 4i – 2j

A: Please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Find the curl of the vector field at the given point. F (x, y, z) = x²zi - 2xzj + yzk, (2,–1,3)…

A: The solution as follows :-

Q: Consider inner-product = | f(x)g(x) dx defined for vector space C[-1, 1] , -1 then for the function…

A:

Q: (b) Given vector fields F and G on R³, show that div(F x G) = Ğ · curl(F) – F - curl(C).

A: Prove that div(F×G)=G.curl(F)-F.curl(G)

Q: Find ||FI|. F(x, y) = yi + xj Sketch several representative vectors in the vector field. y C -2 2 2…

A: given vector fieldF(x, y)=yi+xj

Q: 4. Match the vector fields given below with their plots. (a) F(x,y, z) = (b) F(x, y, z) = (c) F(x,…

A:

Q: Q2) Show that (R², R , +,.) is a vector space or not? Under additional and scalar multiplication on…

A: Vector Space: A set V is called vector space over the scalar field F if two operations + and · are…

Q: span{v1,..., vk}, where {v;}{_1 are orthonormal vectors, be a linear sub- (b) Let V := space of Rd.…

A:

Q: find the scalar projection of v onto w. Also find the vectorprojection of v onto w. v = (2,−3)T , w…

A:

Q: (4) Show that Trab) = a b Tr-Trace (5) Show that T. I = Tr(T) where I is an arbitrary second Order…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: find the scalar projection of v onto w. Also find the vectorprojection of v onto w. v = (4, 1)T , w…

A:

Q: Find the divergence of the vector field V(x, y, z) = –5e*"i – e'0xyj + 4e®y=k. (Give an exact…

A:

Q: b) Plot the vector fields for different values of h (show all qualitative behaviors)

A: The given equation is, dxdτ=x1-x-h So, the vector field for different values of h are as follows.

Q: tch the vector field with its graph. F(x, y) = yi y 事 5 -5- -6+ y 1信 I 5 5 -5+

A: Given  F(x,y)=yi Option (1) is correct by given graph

Q: 2) Using Gauss's theorem also known as divergence theorem, show that f as x P = [(vx F)av, dS ×…

A:

Q: Determine it the set belon defines a vector Supspae. Justify the conclusion All rectors

A:

Q: If aj is a symmetric covariant tensor and b, a cova- . riant vector which satisfy the relation ajjbx…

A:

Q: Let S be the portion of the cylinder x2 + y2 = 1, −1≤z≤1 and F the vector field given by F (x, y, z)…

A:

Q: Determine whether the flux of the vector field F through each surface is positive, negative, or…

A:

Q: Find the divergence of the vector field V(x, y, z) = –5xi + (y + 2 cos(x))j+ (12z + e*x») k. (Give…

A:

Q: Af V s vector space th cV = V

A:

Q: Let ø = p(x), u = u(x), and T = T(x) be differentiable scalar, vector, and tensor fields, where x is…

A: Given:-                   Let ϕ = (x), u = u(x), and T = T(x) be differentiable scalar, vector, and…

Q: Let f, u and S be scalar, vector and (second-order) tensor fields respectively. Using index…

A: The complete solutions are given below

Q: Find the curl and divergence for the given vector field. 1. Posititve 2. Negative div G 3. 0 4. O…

A: The image of a vector field in two dimension is given To determine it's curl and divergence.

Q: Find the curl and divergence for the given vector field. 1. Posititve 2. Negative 3. 0 div G 4. O…

A: The given vector field is:

Q: Let S = {u1, u2, . . . , un} be a linearly independent subset of a vector space V over the field Z2.…

A:

Q: Please answer in complete solution The task is to identify all the values of a and b that will make…

A: