posted 10 months ago

\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|.

posted 10 months ago

\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.

posted 11 months ago

(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 {. }

posted 11 months ago

(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.

posted 11 months ago

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.

posted 11 months ago

(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.

posted 1 years ago

\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}

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

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

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

posted 1 years ago

O_{x}(\theta)=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & -\sin \theta & \cos \theta \end{array}\right)

A rotation around the y-axis by angle ø is given by the matrix

O_{y}(\phi)=\left(\begin{array}{ccc} \cos \phi & 0 & \sin \phi \\ 0 & 1 & 0 \\ -\sin \phi & 0 & \cos \phi \end{array}\right)

and a rotation around the z-axis by angle x is given by the matrix

O_{z}(\chi)=\left(\begin{array}{ccc} \cos \chi & \sin \chi & 0 \\ -\sin \chi & \cos \chi & 0 \\ 0 & 0 & 1 \end{array}\right)

a) The vector V is given by

\mathbf{V}=\left(\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right)

Show by explicit calculations that

O_{z}(\chi=\pi) O_{y}(\phi=\pi / 2) \mathbf{V}=O_{y}(\phi=-\pi / 2) \mathbf{V}

Explain with words the geometric interpretation of this equation.

Evaluate the composite rotation O2(x1)Oz(x2) and express it as a rotation around the z axis with respect to a single angle. By using the iteration procedure, determine0:(x), where n is a positive integer, in terms of a rotation with respect to a single angle.[6]

Find the general angles x and phi of two successive rotations of the form O2(x)O,(phi)acting on vector V such that the resulting vector corresponds to a rotation Ox(0)of V around the x-axis for a given angle phi. Draw the corresponding diagram that depicts the rotations of V with respect to the angles theta, phi and x.[8]

posted 1 years ago

\int_{0}^{1} \frac{x^{4}(1-x)^{4}}{1+x^{2}} d x=\frac{22}{7}-\pi

posted 1 years ago

\text { a. } \int_{1}^{3}\left(\sqrt{\frac{49-x}{x}}-\sqrt{\frac{1 x}{49-x}}\right) d x

\text { b. } \int \frac{5 x^{4}+7 x^{2}+x+2}{x\left(x^{2}+1\right)^{2}} d x

\text { c. } \int \frac{x^{2}}{\left(3+4 x-4 x^{2}\right)^{3 / 2}} d x

\text { d. } \int \frac{8 \cos ^{5} x}{\sqrt{\sin x}} d x