Multivariable Calculus

Vectors

• Magnitude
• Direction

Sample 3d vector notation :

$\vec{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ $\vec{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$

Length of the vector : $|\vec{A}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$

Addition

Sum of two vectors is the diagonal of the parallelogram formed with those two vectors

$\vec{A} + \vec{B} = (a_1 + b_1)\hat{i} + (a_2 + b_2)\hat{j} + (a_3 + b_3)\hat{k}$

Dot Product

It is a scalar

$\vec{A} \centerdot \vec{B} = \sum{a_ib_i}$

Geometrically $\vec{A} \centerdot \vec{B} = |\vec{A}||\vec{B}| cos(\theta)$

Cross Product

It is a vector

$\vec{A} X \vec{B} = det(\vec{A}, \vec{B})$

$\vec{A} X \vec{B}$ signifies the area of the parallelogram formed with those two vectors

Magnitude = $|\vec{A}||\vec{B}| sin(\theta)$ Direction = perpendicular to both the vectors (Right hand thumb rule)

In case of three dimensions $\vec{A} X \vec{B} X \vec{C}$ signifies the volume of paralleopiped formed by A, B, C

Cross product of two vectors in 3d space $\vec{A} X \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

Planes

Given three points in a plane $P_1, P_2, P_3$. Let $P$ is a point in the plane then,

$\vec{P_1P}\centerdot(\vec{P_1P_2}X\vec{P_1P_3}) = 0$

$\equiv det(\vec{P_1P}, \vec{P_1P_2}, \vec{P_1P_3}) = 0$

Equations of planes

$ax + by + cz = d$

Normal vector to the plane : $\langle \hat{a}, \hat{b}, \hat{c} \rangle$

Parametric equations of line

$x(t) = a_1t+b_1, y(t) = a_2t+b_2, z(t) = a_3t+b_3$

Matrices

$AX = B \equiv X = A^{-1}B$

$A^{-1} = adj(A)/det(A)$

In 3D system, in general two planes intersect a line and third plane intersects the line at a point

Other possible solutions are a line, a plane

Rank of a Matrix

Rank of a matrix is number of linearly independent columns or number of linearly independent rows

Trace of a Matrix

Trace of a matrix is sum of its diagonal elements

$trace(A) = \sum_{i}(A_{ii}) = \sum_{i}(\lambda_i)$

Here $\lambda$ is eigen values of the matrix

Inverse of a Matrix

$A^{-1} = Adj(A)/det(A)$

Determinant of a Matrix

$Det(A) = \prod_{i}(\lambda_i)$

Orthogonal Matrix

$AA^T = I = A^TA$

Eigen values and Eigen vectors of a Matrix

For an n x n square matrix A, e is an eigen vector of $A$ with eigen value $\lambda$ if

$Ae = \lambda e \Rightarrow (A - \lambda I)e = 0\Rightarrow det(A - \lambda I) = 0$