By failing to prepare, you are preparing to fail, Benjamin Franklin.
Definition. A vector $\vec{AB}$ is a geometric object or quantity that has both magnitude (or length) and direction. Vectors in an n-dimensional Euclidean space can be represented as coordinates vectors in a Cartesian coordinate system.
Definition. The magnitude of the vector $\vec{A}$, also known as length or norm, is given by the square root of the sum of its components squared, $|\vec{A}|~ or~ ||\vec{A}|| = \sqrt{a_1^2+a_2^2+a_3^2}$, e.g., $||< 3, 2, 1 >|| = \sqrt{3^2+2^2+1^2}=\sqrt{14}$, $||< 3, -4, 5 >|| = \sqrt{3^2+(-4)^2+5^2}=\sqrt{50}=5\sqrt{2}$, or $||< 1, 0, 0 >|| = \sqrt{1^2+0^2+0^2}=\sqrt{1}=1$.
The sum of two vectors is the sum of their components, $\vec{A}+ \vec{B} = (a_1+b_1)\vec{i}+(a_2+b_2)\vec{j}+(a_3+b_3)\vec{k}$ = < (a_{1}+b_{1}), (a_{2}+b_{2}), (a_{3}+b_{3}) >.
The subtraction of two vectors is similar to addition and is also done component-wise, it is given by simply subtracting their corresponding components (x, y, and z in 3D), $\vec{A} - \vec{B} = (a_1-b_1)\vec{i}+(a_2-b_2)\vec{j}+(a_3-b_3)\vec{k}$ = < (a_{1}-b_{1}), (a_{2}-b_{2}), (a_{3}-b_{3}) >.
Scalar multiplication is the multiplication of a vector by a scalar, a real number, changing its magnitude without altering its direction. It is effectively multiplying each component of the vector by the scalar value, $c\vec{A} = (ca_1)\vec{i}+(ca_2)\vec{j}+(ca_3)\vec{k} = < ca_1, ca_2, ca_3>$.
The dot or scalar product is a fundamental operation between two vectors. It produces a scalar quantity that represents the projection of one vector onto another. The dot product is the sum of the products of their corresponding components: $\vec{A}·\vec{B} = \sum a_ib_i = a_1b_1 + a_2b_2 + a_3b_3.$, e.g. $\vec{A}·\vec{B} = \sum a_ib_i = ⟨2, 2, -1⟩·⟨5, -3, 2⟩ = a_1b_1 + a_2b_2 + a_3b_3 = 2·5+2·(-3)+(-1)·2 = 10-6-2 = 2.$
It is the product of their magnitudes multiplied by the cosine of the angle between them, $\vec{A}·\vec{B}=||\vec{A}||·||\vec{B}||·cos(θ).$
The cross product, denoted by $\vec{A}x\vec{B}$, is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both of the input vectors and has a magnitude equal to the area of the parallelogram formed by the two input vectors.
The cross product $\vec{A}x\vec{B}$ can be computed using the following formula, $\vec{A}x\vec{B} = det(\begin{smallmatrix}i & j & k\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\end{smallmatrix}) =|\begin{smallmatrix}a_2 & a_3\\ b_2 & b_3\end{smallmatrix}|\vec{i}-|\begin{smallmatrix}a_1 & a_3\\ b_1 & b_3\end{smallmatrix}|\vec{j}+|\begin{smallmatrix}a_1 & a_2\\ b_1 & b_2\end{smallmatrix}|\vec{k}$
Matrices offers an efficient way for solving systems of linear equations. Utilizing matrices, we can represent these equations more concisely and conveniently.
For a system like,
$(\begin{smallmatrix}2 & 3 & 3\\ 2 & 4 & 4\\ 1 & 1 & 2\end{smallmatrix}) (\begin{smallmatrix}x_1\\ x_2\\x_3\end{smallmatrix}) = (\begin{smallmatrix}u_1\\ u_2\\u_3\end{smallmatrix})$, A · X = U provides a more convenient and concise notation and an efficient way to solve system of linear equations where A is a 3 x 3 matrix representing coefficients, X is a 3 x 1 column vector representing the results where we basically do dot products between the rows of A (a 3 x 3 matrix) and the column vector of X (a 3 x 1 matrix).
Given a square matrix A (it is a matrix that has an equal number of rows and columns), an inverse matrix A^{-1} exists if and only if A is non-singular, meaning its determinant is non-zero (det(A)≠ 0). $A \times A^{-1} = A^{-1} \times A = I$. Essentially, multiplying a matrix by its inverse reverses (“undoes”) the effect of the original matrix.
Consider a system of linear equations represented in matrix form as: AX = B, where A is a n×n matrix (coefficient matrix), X is an n×1 matrix (column vector of variables), and B is an n×1 matrix (column vector of constants).
Solving a system of linear equations expressed as AX = B involves finding or isolating the matrix X when we are given both matrices A and B. AX = B ⇒[A should be non-singular, meaning its determinant is non-zero, det(A)≠0, then we can multiply both side by A^{-1}]⇒ X = A^{-1}B.. $A^{-1} = \frac{1}{\text{det}(A)} \times \text{adj}(A)$. adj(A) = C^{T} where $C_{ij} = (-1)^{i+j} \times \text{minor}(A_{ij})$ and det(A) =[A is a 3x3 matrix] a(ei - fh) - b(di - fg) + c(dh - eg)
Functions of two variables are natural generalizations or extensions of functions of one variable. They are mathematical functions f: ℝ x ℝ → ℝ, meaning they take two real input (x, y) and produce one real number output f(x, y). Essentially, they map each pair of real numbers (x, y) to a third real number f(x,y).
They are fundamental in multivariable calculus and have numerous applications in science, engineering, and economics.
A function of two variables assigns a unique real number to each ordered pair in its domain.
Typically, we express such functions as z = f(x, y), where x and y are the independent variables, and z is the dependent variable.
The domain of a function of two variables is the set of all possible input values for both variables x and y. Here are some examples:
The graph of a two-variable function is typically a surface in a three-dimensional Cartesian space (Figure v). This surface represents all points (x, y, z) such that z = f(x, y).
To illustrate, consider the function f(x, y) = x^{2} + y^{2}.
Here’s how you can visualize it:
The graph is a paraboloid, opening upwards. At (0, 0), z is at its minimum value, 0. As you move away from the origin in any direction on the xy-plane, z increases quadratically.
f(x, y) = -y, Figure vi, the graph is a plane that slopes downwards along the positive y-axis and upwards the negative y-axis. Although the picture may not be clear in three dimensions, the function passes through points like (0, 1, −1) and (0, -1, 1), indicating that at y = 1, z is −1, and at y = −1, z is 1.
f(x, y) = 1 -x^{2} -y^{2} (Figure vii). Notice that in the yz plane, where x = 0, the graph is a parabola z = 1 -y^{2}. Similarly, in the xz plane, where y = 0, the graph is a parabola z = 1 -x^{2} and in the xy plane, where z = 0, 1 = y^{2} +x^{2} forms the unit circle. The graph is a paraboloid opening downward.
f(x, y) = $\sqrt{4-x^2-y^2}↭[\text{Rewrite in Terms of z}] z^2 = 4 -x^2 -y^2 ↭ x^2 +y^2 + z^2 = 4$. This equation represents a sphere centered at the origin with a radius of 2. However, since z = f(x, y) = $\sqrt{4-x^2-y^2}$ is non-negative, the graph represents the upper hemisphere (Illustration 2).
f(x, y) = $1-x-\frac{1}{2}y$ is a plane, a flat surface that extends infinitely in all directions. The easiest way to plot a plane is to find the x-, y-, and z-intercepts, namely (0, 0, 1), (0, 2, 0), (1, 0, 0) (Illustration 3).
Definition. Contour plots are used to represent three-dimensional data on a two-dimensional surface. They are similar to topographical maps, where instead of plotting longitude, latitude, and elevation, we plot x-, y-, and z-values..
A contour plot displays level curves of a function z = f(x, y), i.e., it shows all the points where z = f(x, y) equals some fixed constant, aiding in the visualization of the function’s gradient and overall shape, e.g., z = x^{2} + y^{2} (Illustration 4), Figure via, Figure ia, Figure viia.
A level curve is simply a cross section of the graph of z = f(x,y) taken at a constant value, say z = c. A level curve can be interpreted as a contour line on a map, where each line represents points with the same elevation.
Creating a Contour Plot:
If the level curves are getting closer and closer, the function is getting steeper and steeper (picture it like a mountain)