MY-ASSIGNMENTEXPERT™可以为您提供sydney MATH3078 Partial Differential Equations偏微分方程的代写代考和辅导服务！
这是悉尼大学 偏微分方程的代写成功案例。
MATH3078课程简介
The aim of this unit is to introduce some fundamental concepts of the theory of partial differential equations (PDEs) arising in Physics, Chemistry, Biology and Mathematical Finance. The focus is mainly on linear equations but some important examples of nonlinear equations and related phenomena re introduced as well. After an introductory lecture, we proceed with first-order PDEs and the method of characteristics. Here, we also nonlinear transport equations and shock waves are discussed. Then the theory of the elliptic equations is presented with an emphasis on eigenvalue problems and their application to solve parabolic and hyperbolic initial boundary-value problems. The Maximum principle and Harnack’s inequality will be discussed and the theory of Green’s functions.
Prerequisites
At the completion of this unit, you should be able to:
- LO1. demonstrating the ability to recognize different types of partial differential equations: “linear” or “nonlinear”, “order of the given equation”, “homogeneous” or “inhomogeneous”, and if it concerns 2nd-order equations, whether they are of “elliptic”, “parabolic”, or “hyperbolic” type
- LO2. demonstrating the conceptional understanding of how to apply different methods for solving different types of partial differential equations. Those methods include the use of classical ODE-concepts to solve PDEs
- LO3. understanding the definitions, main theorem, and corollaries of Green’s functions and Poisson kernel
- LO4. be fluent with “change of variable” into polar, cylindrical and spherically coordinates and to be able to compute partial derivatives in these coordinates
- LO5. develop an appreciation and strong working knowledge of the theory and application of elementary partial differential equations
- LO6. be fluent in using generalized Fourier transforms to solve parabolic and hyperbolic initial boundary value problems where the spatial variable might be of more than one variable
MATH3078 Partial Differential Equations HELP（EXAM HELP， ONLINE TUTOR）
a) Find the matrix of a rotation with dilation which rotates in the plane around the origin by $45^{\circ}$ counterclockwise and scales by a factor $\sqrt{2}$.
b) Find the matrix of rotation with dilation which rotates around the origin by $60^{\circ}$ counterclockwise and scales by a factor 2 .
Name the following transformations. Choose from Dilation, Shear, Rotation, Projection, Reflection, and Rotation with Dilation. (Note: not all transformations might occur.)
a) $A=\left[\begin{array}{ll}0 & 0 \ 0 & 1\end{array}\right]$, b) $A=\left[\begin{array}{cc}3 & -2 \ 2 & 3\end{array}\right]$, c) $A=\left[\begin{array}{cc}1 & -2 \ 0 & 1\end{array}\right]$,
d) $A=\left[\begin{array}{ll}0 & 1 \ 1 & 0\end{array}\right]$
e) $A=\left[\begin{array}{cc}1 & 0 \ -2 & 1\end{array}\right]$
f) $A=\left[\begin{array}{cc}0 & 1 \ -1 & 0\end{array}\right]$
Pair the linear transformations given by the matrices $A, B, C, D, E$ below with one of the following types:
a) an orthogonal projection onto a line
b) reflection about a line
c) a rotation around a line
d) a reflection about a plane
e) an orthogonal projection onto a plane
$$
\begin{aligned}
& A=\frac{1}{9}\left[\begin{array}{ccc}
1 & -2 & 2 \
-2 & 4 & -4 \
2 & -4 & 4 \
2 & -2 & -1 \
-2 & -1 & -2 \
-1 & -2 & 2
\end{array}\right], B=\frac{1}{9}\left[\begin{array}{ccc}
5 & -4 & -2 \
-4 & 5 & -2 \
-2 & -2 & 8
\end{array}\right], C=\frac{1}{3}\left[\begin{array}{ccc}
-2 & 2 & 1 \
2 & 1 & 2 \
1 & 2 & -2
\end{array}\right], \
& \left.D=\frac{1}{3} \begin{array}{ccc}
0 & 5 & 0 \
-4 & 0 & 3
\end{array}\right] .
\end{aligned}
$$
Assume $a^2+b^2=1$. One of the three transformations is a rotation, the other is a reflection about a line, the third is an orthogonal projection onto a line. Which is which? Find the inverse in the case of rotation and reflection.
a) $A=\left[\begin{array}{cc}a & -b \ b & a\end{array}\right]$.
b) $B=\left[\begin{array}{ll}a^2 & a b \ a b & b^2\end{array}\right]$.
c) $C=\left[\begin{array}{cc}a & b \ b & -a\end{array}\right]$.
For a $2 \times 2$ matrix $A=\left[\begin{array}{ll}a & b \ c & d\end{array}\right]$, the determinant is $\operatorname{det} A=a d-b c$. Find the determinant of a shear, of a rotation, of a reflection about a line, of reflection in the origin, a projection onto the $x$ axis, a rotation-dilation matrix with parameters $(a, b)$.
MY-ASSIGNMENTEXPERT™可以为您提供SYDNEY MATH3078 PARTIAL DIFFERENTIAL EQUATIONS偏微分方程的代写代考和辅导服务！
