Use the inner product 1 (f, 8) = f(x)g(x) dx in the vector space CO[0, 1] of continuous functions on the domain [0, 1] to find (f, g), ||f ||, ||g|| for f(x) = -5x² – 6 and g(x) = 6x + 1. %D (f,g) = ||S|| = lg| :

Q: Prove that gradient of vector potential is zero

A: See the attachment

Q: Define left singular vectors.

A: To Define:Left singular vectors.

Q: Tensor Auv is type (2 0) and tensor Wv is type (0 1). prove that AuvWv is a tensor of type (1 0)…

A: It is given that, Auv be tensor of type 2,0. Wv be tensor of type 0,1. it is known to us that if A…

Q: let (x,y,z) be Scalar Field and F a vector field.. $ Derive expression for D. (OF) and * (OF) in…

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: How do you find the length of a space curve given by a vector function r(t)

A:

Q: Determine whether the statement" The weight of a car is a scalar", is true or false. If it is false,…

A: Scalar: Many quantities in geometry and physics, such as volume, mass, temperature and time can be…

Q: Determine if the scalar vector field is Continuous on the indicated domain f(x) = (x²+1) S=(0,0) Х…

A: f(x)=x2+1/x​ is not Lipschitz continuous on the indicated domain.Explanation:Approach to solving the…

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: Find the divergence of the vector field F = div F =

A:

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

A: We have to show .

Q: How do dual spaces in vector spaces influence the understanding of functional analysis

A: The concept of dual spaces plays a fundamental role in functional analysis and significantly…

Q: Find the parametric form of the equation of the plane passing through the points (1, 3, −1), (2, 2,…

A: Points :1, 3, -1, 2, 2, 3, and -1, 1, 1

Q: gradient of scaler function obeys the vector transformation law

A: Given that the function is scalar. So f(x,y,z)=C Where C is function of x, y, z. So the gradient of…

Q: Vector A satisfies the vector transformation law, A'=SA. Show directly that its time derivative…

A: Given that a vector A satisfies the vector transformation A'=SA Query is to show that the derivative…

Q: Define the terms sink and source in vector fields.

A:

Q: True or False? The set {0,1} along with the usual definitions of addition and scalar multiplication…

A:

Q: Determine if the scalar vector field is Lipschitz continuous on the indicated domain ON = f(x) = tan…

A: The detailed solution is provided below.Explanation:Given, f(x)=tanx,Ω=(−2π​,2π​) f′(x)=sec2(x)…

Q: Sketch an example of a nonconstant vector field in the plane inwhich each vector is parallel to (1,…

A: We have to sketch an example of a non constant vector field in the plane in which each vector is…

Q: Determine if the scalar vector field is Lipschitz continuous on the indicated domain f(x) = ex n = R

A: f(x)=ex is not lipschitz continuous in  Ω=RExplanation:Suppose f(x)=ex is Lipschitz on R.  Then,…

Q: Identify whether the given vector field has positive, zero or negative divergence. Zero Postive…

A: We need to determine if the divergence of the given vector field is positive, zero or negative. The…

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: give two example of tangent vector on manifold

A: Manifold is a topological space that which is almost similar to a euclidean space near to each of…

Q: Write formulas for the unit normal and binormal vectors of a smooth space curve r(t)

A: The formulas for unit normal vector  and binomial vectors of a smooth space curve r(t).

Q: Find the divergence of the vector field F - div F = =

A:

Q: Determine if the scalar vector field is Lip schit Continuous on the indicated domain f(x) = ex, 2 =…

A: As f(x) = ex satisfies the Lipschitz condition with Lipschitz constant K = M on the domain (−∞,0).…

Q: grad (фu) —D и @ grad ф + ф grad u

A:

Q: Find the divergence of the vector field F =< yx³, xzª, zy² div F = V

A: Given : F→=&lt;yx3,xz4,zy2&gt;   To Find : Divergence of given vector field

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

A: The solution is given as

Q: V = M ₂x2 (IR) let M = { A€ V A = A} N = {A = V A = - A^²} Subset of the vector space V The answer V…

A: To prove  V=M⊕N we need to prove  (i) V=M+N and  (ii) M∩N=0

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.