• (2,1,1) 20. (2x In y – yz) dx + хz ) dy — ху аdz - у (1,2,1)

Q: find the work done by each field along thepaths from (0, 0, 0) to (1, 1, 1) F = 2xy i + x2 j + k

A: find the work done by each field along thepaths from (0, 0, 0) to (1, 1, 1) F = 2xy i + x2 j + k

Q: 3. Let W = M2x2(C) with zero determinant. Show that W is not a subspace of V = M2x2(C).

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Q: 1 1 -2) 13. Find the rank of A=| 2 1 |and explain why it indicates that A is (-1 -1 2 not…

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Q: 2/7,545

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Q: Show that X'= B(X, K) is a Banach space.

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Q: If (0,1) ~ (-1,1) and (-1,1) ~ R. Deduce that (0,1) ~R

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Q: 3. Use Gauss' theorem to evaluate IL (* + y + 2) do, where E is the surface of the ball x? + y? + z²…

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Q: be a field and let c,d e F. Show that c · (-d) = -(c d).

A: Associative Property of Field F for a,b,c∈F a·b·c=a·b·c

Q: Let f and g be any two smooth functions (defined on the same domain in R*), and let V = Vf and W =…

A: The given statement is

Q: Evaluate the work done between point 1 and point 2 for the conservative field F. F 3D6xi+ 6уј+ 6z k;…

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Q: Let F be an inverse square field, that is, F(r) = cr/|r|3 for some constant c, where r = xi + yj +…

A: Given fields is F(r)=crr3 Flux of the field F will be,∬SF.dS=∬SF.ndShere n is the unit normal…

Q: suppose cX) is valued function on such that f?E R Faib] Does it fallows bounded real taib] that…

A: In the given question given that f(x) is a bounded real valued function on [a,b] then we have to…

Q: Let f(2)= ze² (z - Ti) 4 Verify that mi is a pole of f of order 4. Determine Res f(z). 2=Ti

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Q: se the given transformation to evaluate the integral. J 6z + 3ydA, where Ris the parallelogram with…

A: we have given: x=13u+v and y=132u-v  u=x+y, v=2x-y Now, Consider the integrals, ∬R6x+3ydA where, R…

Q: Use the field F =Z2[x]/<x2 + x + 1> to construct a (5, 2) linearcode that will correct any…

A: Consider the given field, F=Z2x/x2+x+1 Now, abbreviate the coset a+x2+x+1 with a. Then the…

Q: Which elements are invertible in Z4[x,y]?

A: W need to find elements which are invertible inZ4x,y.

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: 2. True/False: If F, G are vector fields and V × F = V × G, then F = G. (Justify your answer by…

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Q: Let fG) = x2+I E Z, x] and let R= Z, [x]/I, where I= (f (x)) @ Show that R is a field with 9…

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Q: 1. This question concerns the field F(x, y) = (2, -x). a) Plot field F in the window [-3, 3] x…

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Q: Test the field f to determine if it's conservative. F=x⁵y⁵z⁵i+x⁵y⁵z⁵j+z⁵k Conservative or not…

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Q: Is = xjx + yjy+zźz3 an inner product on R'?

A: No it is not an inner product.  Inner product looks like a dot product.

Q: When multiplying a scalar and a vector: |xa|=A|ā| when A |xa|=|||ā| when X + 0 and ◆ 0.

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Q: (Chapter 17,Section 19.1) Consider the region R bounded by the lines y = −3x and Y = 3x and the…

A: Consider the region R bounded by the lines .To Find:a) 2-dimensional vector field  such that .b)…

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: Let W be the work (against the Sun’s gravitational force) required to transport an 80-kg person from…

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Q: Prob 2. Let TEL(V). Show that (v, u)r: (Tv, u) is an inner product on V if and only if T is positive…

A: Introduction: An invertible positive operator is a linear operator on a vector space that is both…

Q: R^n×n is invertible, show that AB is similar to BA for all B ∈ R^n×n.

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Q: Proved that the set V of all real valued functions is a vector space under the operations defined in…

A: Here in the question, it is given that V is a collection of all continuous functions. So, then it…

Q: Prove that the vector space C of all continuous functions from R to R is infinite-dimensional. Shew…

A: We have to prove given statement:

Q: If φ and ψ are smooth scalar fields, show that ∇×(φ∇ψ)=∇φ×∇ψ

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Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: