Solved: A vector field V(r) is given in cylindrical coordinates by

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a vector field vr is given in cylindrical coordinates by mathbfvmathbf

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Vectors

A vector field V(r) is given in cylindrical coordinates by

\mathbf{V}(\mathbf{r})=\rho^{2} \hat{\boldsymbol{\rho}}+\frac{z^{2}}{\rho} \cos \phi \hat{\boldsymbol{\phi}}+2 z \sin \phi \hat{\mathbf{z}}

Calculate V . V and ▼ × V.

Determine if the vector field V is (i) solenoidal, (ii) irrotational (iii) conservative in the whole space and explain your answer.[5]

) A path C starts from the point W = (1,0, 0), to the point X = (1,0, 1) along thestraight line x = 1, y = 0 and z from 0 to 1, then from X to Y = (-1,0, 1) alongthe semicircle with z = 1, p= 1 and ø ranging from 0 to 7 anticlockwise, then tothe point Z = (-1,0,0) along the straight line x = -1, y = 0 and z from 1 to0, and finally from Z to W along the semicircle with p= 1, z = 0 and ø from Ato 0 clockwise. Draw the path C in Cartesian coordinates and explicitly calculatethe line integral

I_{C}=\oint_{C} \mathbf{v} \cdot d \mathbf{r}

explaining all your steps.

Calculate the surface integral

I_{S}=\int_{S} \nabla \times \mathbf{V} \cdot d \mathbf{S}

where S is the surface on the curved part of the cylinder enclosed by the loop C.Verify with your explicit calculations of Ic and Is the Stokes' theorem given by

\int_{S}(\boldsymbol{\nabla} \times \mathbf{V}) \cdot d \mathbf{S}=\oint_{C} \mathbf{V} \cdot d \mathbf{r}

\text { Find the function } f(\rho, \phi, z) \text { such that } \mathbf{V}=\boldsymbol{\nabla} f \text { with } f(0,0,0)=0 \text {. }

Answer

Verified

Question 45499

Vectors

Suppose that
\mathbf{A}=\left(\begin{array}{cccc} 1 & x & y & 1 \\ 1 & x & x & x \\ x & 1 & x y & y \\ x & x & x y & 1 \end{array}\right) \in \mathscr{M}_{4,4}(\mathbb{R})
Show that A is not invertible if and only if either x = y, x =1 or xy = 1.
Let V and W be 3-dimensional vector spaces over R, Bị = {V1, V2, V3} be an ordered basis for V and B2 = {w1,w2, w3} be an ordered basis for W. Define a linear transformation T :V → W by
T\left(\mathbf{v}_{1}\right)=\mathbf{w}_{1}+\mathbf{w}_{2}
T\left(\mathbf{v}_{2}\right)=\mathbf{w}_{1}-\mathbf{w}_{2}+\mathbf{w}_{3}
T\left(\mathbf{v}_{3}\right)=\mathbf{w}_{3}
(i) Write down the matrix A representing T with respect to the ordered bases B1 and
(ii) Calculate the dimension of the kernel of T and of the image of T.
(iii) Show that B3 = {v1,V1 + V2, V1 + V2 + V3 } is a basis for V and B4 = {w1 -
W2, W2 +W3, W1 – W3} is a basis for W.
) Find the matrix which represents T with respect to the ordered bases B3 and B4.

Vectors

Prove that
\delta\left[a\left(x-x_{1}\right)\right]=\frac{1}{a} \delta\left(x-x_{1}\right) .
\text { Note. If } \delta\left[a\left(x-x_{1}\right)\right] \text { is considered even, relative to } x_{1} \text {, the relation holds for negative } a
and 1/a may be replaced by 1/la|.

Vectors

Problem 6. (Exercise 6 in the lecture on6.23) Finish the proof of either of the following statements:
(a) Let S =then S contains a subset S', such that S' is a basis ofV.{v1, v2, ..., Um} be a spanning set of a finite dimensional F-vector space V,
(b) Let S = {v1, V2, . .. , Um} be a linearly independent set of a finite dimensional F-vector space V, then S is a subset of some S' C V, such that S' is a basis of V.
Using the statement that you proved, show that if W is a subspace of V, then dim W <dim V.

Vectors

Problem 5. (Exercise 5 in the lecture on 6.23) Let V be an F-vector space, {v1,..., Vm} be a list of vectors in V. Show that the two definitions of linear independence are equivalent:
(a) For any i = 1,..., m, v; is not a linear combination of the rest of the vectors in the list.
\text { (b) Let } a_{1}, \ldots, a_{m} \in \mathbb{F} \text {, then } a_{1} v_{1}+a_{2} v_{2}+\ldots+a_{m} v_{m}=0 \text { if and only if } a_{1}=\ldots=a_{m}=0 \text {. }

Vectors

Problem 3. (Exercise 4 in the lecture on 6.23) Let F be a field and Fm[r] be the set ofpolynomials with coefficients in F whose degree is less than m. Show that Fm[x], equippedwith the usual addition and scalar multiplication, is a vector space over F. In the lecture on6.23, we will show that F[r], the set of polynomials of arbitrary degrees with coefficients in F,is a vector space. So you only need to show that Fm[r] is a subspace of F[r] for any positiveinteger m.
Problem 3. (Exercise 4 in the lecture on 6.23) Let F be a field and Fm[z] be the set ofpolynomials with coefficients in F whose degree is less than m. Show that Fm[r], equippedwith the usual addition and scalar multiplication, is a vector space over F. In the lecture on6.23, we will show that F[r], the set of polynomials of arbitrary degrees with coefficients in F,is a vector space. So you only need to show that Fm[r] is a subspace of F[r] for any positiveinteger m.

Vectors

Problem 1. (Exercise 1 in the lecture onfield:6.21) Using the definition of fields, show that in a
(a) The zero scalar is unique,
(b) The additive inverse of any scalar is unique,
(c) The unit scalar is unique,
(d) The multiplicative inverse of any nonzero scalar is unique.

Vectors

k + 0 is an eigenvalue with corresponding eigenvector (1, 2, 3) of
A=\left[\begin{array}{ccc} -6 & -2 & 3 \\ c & a & 4 \\ -3 & -6 & b \end{array}\right]
Find a, b, k.

Vectors

\text { For the matrix } A=\left[\begin{array}{rrr} 4 & 8 & 0 \\ -1 & -2 & 0 \\ 0 & 0 & 2 \end{array}\right] \text { find all the eigenvalues and for each }
eigenvalue the general solution to the eigenvector problem.

Vectors

\text { a) The matrix } Y=\left[\begin{array}{lll} 0 & 1 & 2 \\ 0 & 0 & 4 \\ 0 & 0 & 0 \end{array}\right] \text { has } 0 \text { as its only eigenvalue. }
Find the general solution to the problem of finding the corresponding eigenvectors.
\text { The matrix } Z=\left[\begin{array}{lll} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{array}\right] \text { has } 1 \text { as its only eigenvalue }
Find the general solution to the problem of finding the corresponding eigenvectors.

Vectors

\text { For the matrix } X=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right] \text { find the eigenvalues and an eigenvector }
corresponding to each.