An introduction to using the quotient rule

Find the dot product and the angle between two vectors

Find the cosine of the angle between two pairs of 3D and 4D vectors.

Some questions to practice your fundamental skills in vectors.

Several quadratics are given and students are asked to complete the square.

Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$.

$A,\;B$ $2 \times 2$ matrices. Find eigenvalues and eigenvectors of both. Hence or otherwise, find $B^n$ for largish $n$.

Find the determinant of a $3 \times 3$ matrix.

$A$ a $3 \times 3$ matrix. Using row operations on the augmented matrix $\left(A | I_3\right)$ reduce to $\left(I_3 | A^{-1}\right)$.

When are vectors $\boldsymbol{v,\;w}$ orthogonal?

Given vectors $\boldsymbol{v}$ and $\boldsymbol{w}$, find their inner product.

Given vectors $\boldsymbol{v}$ and $\boldsymbol{w}$, find their inner product.

Find the solution of a first order separable differential equation of the form $a\sin(x)y'=by\cos(x)$.

Determine if various combinations of vectors are defined or not.

Elementary operations on vectors; sum, modulus, unit vector, scalar multiple. 

Find the unit vector parallel to a given vector.

Directional derivative of a scalar field.

Curl and divergence of a vector field.  Determine whether the vector field is irrotational or solenoidal.

Divergence of vector fields.

Solve: $\displaystyle \frac{d^2y}{dx^2}+2a\frac{dy}{dx}+(a^2+b^2)y=0,\;y(0)=1$ and $y'(0)=c$. 

Solve: $\displaystyle \frac{d^2y}{dx^2}+2a\frac{dy}{dx}+a^2y=0,\;y(0)=c$ and $y(1)=d$.  (Equal roots example).

Solve for $x(t)$, $\displaystyle\frac{dx}{dt}=\frac{a}{(x+b)^n},\;x(0)=0$

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

Gradient of $f(x,y,z)$.

Find the stationary points of the function: $f(x,y)=a x ^ 3 + b x ^ 2 y + c y ^ 2 x + dy$ by choosing from a list of points.

Gradient of $f(x,y,z)$.

Chain rule

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An introduction to using the quotient rule

Chain rule