Fun with Math!

Sunday, May 31, 2020

Level Surface

If a function w = f (x,y,z) is assigned to a value c, i.e. f (x,y,z) = c, then the surface f (x,y,z) = c is named as the level surface of f.

There are two functions given in the box below: f (x,y,z) and g(x,y,z). By default, f (x,y,z) = 1 is shown. Tick on other choices of c from the same function to observe the changes on the level suface. You may click here if you want to rotate the graph.

Betty, created with GeoGebra and Thomas' Calculus, Pearson.

Level Curve

If a function z = f (x,y) is assigned to a value c, i.e. f (x,y) = c, then the curve f (x,y) = c is named as the level curve of f.

If you take a look at the graph below, there is a paraboloid z = 10 - x² - y² at the right. By default, the choice of 'Show level curve z = 6' is ticked. The surface is 'cut' through by the plane z = 6 and the intersection yields the circle x² + y² = 4.

$\begin{align*} z=f(x,y) &= 10-x^2-y^2\\ 6&=10-x^2-y^2\\ x^2+y^2 &=4\end{align*}$
The circle is then drawn on the xy-plane at both columns. The circle on the xy-plane is the level curve.

Choose different level of c and observe the changes on level curves. Click here if you want to rotate the 3D graph.

Betty, created with GeoGebra and Thomas' Calculus, Pearson.

Friday, May 29, 2020

Domain & Range

Domain of a function is the largest possible set of input to make the function defined. The range is the interval of the function value where the function is defined. If the function is y = f (x), the biggest set of x values is the domain and the interval of y values is the range; if the function is z = f (x,y), the biggest combination set of x and y values is the domain and the interval of the output, z values, is the range.

The graphs below give some examples for better understanding. Please just choose one function at a time to avoid overlapping of domain and/or range of the functions.

A) Function of one variable

The line in purple on the x-axis indicate the interval of the domain, whereas the red line on y-axis indicate the interval of the range. Take note of the domain and range stated in the upper box so that you know how to write it by yourself.

B) Functions of several variables

The graph of the domain is shown at the lower left box and the graph of the function is shown at the right. The interval of the range is marked in the graph at right with light green line. Please click here if you want to have better viewing experience for 3D graphs. Take note of the domain and range stated in the upper box, it is slightly different with the writing of domain as in part A) since the functions involved are functions of more than one variable.

Betty, created with GeoGebra and Thomas' Calculus, Pearson.

Wednesday, May 27, 2020

Tangent & Normal Vectors in Motion

If a particle moves acording to r(t), unit tangent T(t) and principal unit normal N(t) tells about its movement direction. From the previous post, we know that T(t) is the unit vector of velocity, that means, v(t) can be written in term of T(t) as

v(t) = |v(t)|T(t).

Not only v(t) can be written in term of T(t), its derivative - acceleration a(t) can be written in terms of T(t) and N(t) as well:

a(t) = a_TT + a_NN
where a_T is named as the tangential component and a_N as normal component of acceleration. Both of these components can be found using the formulae below:

$a_T=\frac{d}{dt}|\textbf{v}(t)|$
$\begin{align*} a_N &= \sqrt{|\textbf{a}(t)|^2-a_T^2}\\ &= \kappa|\textbf{v}(t)|^2 \end{align*}$

There are two formulae to find a_N as given above. Use either one that is convenient to you.

Using the example below, you can see that the red circle is the curve for a(t). It can be obtained through adding a_TT and a_NN. Slide the value of t if you want to move point C on the circle. If you untick the curve a(t) and tick for v(t), you will find a curve for v(t) and the curve is obtained through the vector |v(t)|T(t) from the origin to point B. Slide the value of t if you want to move point B on the cirlce. Untick v(t) and tick for r(t) now, this curve is the curve of position vector r(t). You can see that the unit tangent T(t) and its principal unit normal N(t) at point A.

If you tick all curves, you will find that all vectors of T(t), |v(t)|T(t) and a_TT are parallel, only differs in length. Same goes to N(t) and a_NN. Any other interesting stuff you can find from this graph? If you want to rotate the graph, just click here.

Betty, created with GeoGebra and Thomas' Calculus, Pearson.

Sunday, May 24, 2020

Motion in Space

If a particle is moving according to the path r(t) (it's also named as position vector), then its velocity, v(t) is found by differentiating r(t) and its acceleration is by differentiating v(t):

$\begin{align*} \textbf{v}(t) &= \frac{\mathrm{d\textbf{r}} }{\mathrm{d} t}\\ \textbf{a}(t) &= \frac{\mathrm{d\textbf{v}} }{\mathrm{d} t}\\ &= \frac{\mathrm{d}}{\mathrm{d} t}\left ( \frac{\mathrm{d\textbf{r}} }{\mathrm{d} t} \right )\\ &= \frac{\mathrm{d^2\textbf{r}} }{\mathrm{d} t^2} \end{align*}$

The unit vector of the velocity has a special name: unit tangent, T. As we know, tangent line is the line that gives us an idea about the slope of a curve, similar to tangent vector.

$\textbf{T}(t)= \frac{\textbf{v}(t)}{|\textbf{v}(t)|}$
In addition to this, there is a principal unit normal, N, that points to the direction to where the unit tangent T is turning.

$\textbf{N}(t)= \frac{\textbf{T}'(t)}{|\textbf{T}'(t)|}$
In order to calculate the rate at which 𝐓 turns per unit of length along the curve, we look for curvature, 𝜅 (“kappa”).

$\kappa(t)= \frac{|\textbf{T}'(t)|}{|\textbf{v}(t)|}$
The graph below gives an example for the calculation mentioned above. Slide the value of t in the upper box to move point A along the curve. Observe the changes on the unit tangent and the pricipal unit normal. In this case, the curvature remain constant throughout all values of t. You can always click here if you want to rotate the graph.

For every curve, there exists a circle of curvature with radius $\rho=\frac{1}{\kappa}$ . This circle of radius satisfies the following critiria:

It is tangential to the respective curve.
It has the same curvature as the respective.
It lies towards the concave or the inner side of the curve.

The green curve in the graph below is the graph of $\textbf{r}(t)=<t^2,t>$ . The blue line is the tangent line of the curve, and the red circle is the circle of curvature. Try to slide the value of t to see the changes of the radius. You can get the same effect by moving point A.

Betty, created with GeoGebra and Thomas' Calculus, Pearson.

Saturday, May 23, 2020

Cylinders

A cylinder in Calculus is not only a daily life cylinder that we know. It is a surface that is generated by moving a straight line along a given planar curve while holding the line parallel to a given fixed line.The curve is called a generating curve.

In other word, a 2D graph is extended. For example, the curve z = y² in yz-plane is 'extended' to all values of x. The box below contains 3 different types of cylinders. Each cylinder is extended to different values of x, y or z. Take note of the corresponding 'base' at the right. According to the example z = y² (Cylinder 3), its 'base' is on yz-plane (check with the axes labels), then the 'base' is 'extended' to all values of x. Slide the value a, b or c to play around with the graph. If you wish to rotate the 3D graph, just click here.

Betty, created with GeoGebra and Thomas' Calculus, Pearson.

Friday, May 22, 2020

Length of a Curve

If a curve is given as
$\textbf{r}(t)=<x(t),y(t),z(t)>$ , then the length of this curve for a < t < b is

$L=\int_a^b\: \sqrt{\left ( \frac{dx}{dt} \right )^2+\left ( \frac{dy}{dt} \right )^2+\left ( \frac{dz}{dt} \right )^2}\;\:dt$ .

For example, given a curve r(t) = < 5cos(t), 5sin(t), t >. The length of the curve for a < t < b is

$\begin{align*}L&=\int_a^b\: \sqrt{[-5\sin(t)]^2+[5\cos(t)]^2+1^2}\;dt\\ &=\int_a^b\: \sqrt{25\sin^2(t)+25\cos^2(t)+1}\;\:dt\\ &=\int_a^b\: \sqrt{25+1}\;\:dt\\ &=\left [ t\sqrt{26} \right ]_a^b\\ &=(b-a)\sqrt{26}\end{align*}$
The graph of the curve mentioned above is shown below (Curve 1). Slide the values of a and b and observe the change in the length. The correspoding segment is indicated in the graph. Let's try to find the length of Curve 2 by yourself and check if you get the same answer as shown in the box below. If you wish to rotate the graph, just click here.

Betty, created with GeoGebra and Thomas' Calculus, Pearson.

Wednesday, May 20, 2020

Volume of Parallelepiped

We already knew that the volume of a box = area x height, but a slanted box (parallelepiped) like the picture below? We may still use the concept of area x height to find the volume. But how? First, assign three vectors u, v and w that start from the same point as in the graph below. We can see that points A and B are both fall on the xy-plane. Considering that t could be an obtuse angle, absolute sign is put in when calculating the height. As such, we get

$\begin{align*} \textrm{Area} &=|\textbf{u}\times \textbf{v}|\\ \textrm{Height}&=|\textbf{w}||\cos t|\\ \textrm{Volume}&=|\textbf{u}\times \textbf{v}||\textbf{w}||\cos t|\\ &=|(\textbf{u}\times \textbf{v})\cdot\textbf{w}|\end{align*}$

$(\textbf{u}\times \textbf{v})\cdot\textbf{w}$ is also named as the box product. Besides finding the cross product, followed by the dot product, you may combine the three vectors into a 3-by-3 matrix and use the determinant method to find the box product (refer in the box below).

In the graph below, move the points A, B or C and observe the change to the respestive vector, together with the change in the volume.

Betty, created with GeoGebra and Thomas Calculus, Pearson.

Tuesday, May 19, 2020

General Curves

A curve in 2D or 3D can be descibed by vector function. So, what is vector function? If vector is a set of numbers, then vector fuction is a set of functions.

In 2D, v(t) = < x(t), y(t) >; in 3D, v(t) = < x(t), y(t), z(t) >.

For examples, the graph below contains 3 curves:

r(t) = < t, t² >
r(t) = < cos(t), sin(t) >
r(t) = < cos(t), t >

Tick one of the vector functions in the graph, and slide the value of t. Observe the position of the terminal point of the vector v(t), P(x, y). As t changes, point P changes according to the vector. If the trace of point P is connected with a smooth curve, we get the following equations:

No.	Components	Relationship between x and y
1.	$\begin{align} x &= t\\ y &= t^2 \end{align}$	$y=x^2$
2.	$\begin{align} x &= \cos(t)\\ y &= \sin(t) \end{align}$	$\begin{align} \cos^2(t)+\sin^2(t) &= 1\\ x^2+y^2 &= 1 \end{align}$
3.	$\begin{align} x &= \cos(t)\\ y &= t \end{align}$	$x=\cos(y)$

In 3D, we don't put the variables in just one equations as in 2D. We leave it as it is in parametric equations. You may slide the value of t below and observe the forming of the curve in 3D (Click here if you wish to have better viewing experience). Even though it is not easy to relate the variables as in 2D, we may still able to predict the shape of some curves. For examples, the graph below contains 3 vector functions:

r(t) = < 3cos(t), 3sin(t), t >, this gives to a circular curve of radius 3 units. As t increases, z increases as well. As such, point P forms a spring of radius 3 units.
r(t) = < cos(t), sin(t), sin(2t) >, since z is not simply t as in #1, we can only expect a curve that fluctuate between [0, 1] for x, y and z.
r(t) = < cos(3t), sin(3t), t >, this looks similar to #1, but with smaller radius (radius = 1 unit) and more intense.

Betty, created with GeoGebra and Thomas Calculus, Pearson.

Friday, May 15, 2020

Quadric Surfaces

A quadric surface is a surface defined by second-degree equations in 𝑥,𝑦 and 𝑧.

$Ax^2+By^2+Cz^2+Dx+Ey+Fz+G=0$

Examples of quadric surfaces including

Ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$
Elliptical paraboloid $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z}{c}$
Elliptical cones $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z^2}{c^2}$
Hyperboloid of one sheet $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$
Hyperboloid of two sheets $-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$
Hyperbolic paraboloid $-\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z}{c}$

Click on the names to find out more about them.

Betty, created with GeoGebra and Thomas Calculus, Pearson.

Hyperbolic Paraboloid

A hyperbolic paraboloid is a type of quadric surfaces. If it is centered at the origin, then its equation could be one of the equations below:

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{z}{c}\;,\;\;a,b,c\neq 0$
$\frac{x^2}{a^2}-\frac{z^2}{c^2}=\frac{y}{b}\;,\;\;a,b,c\neq 0$
$\frac{y^2}{b^2}-\frac{z^2}{c^2}=\frac{x}{a}\;,\;\;a,b,c\neq 0$
Hyperbolic paraboloid has a very interesting shape. It is a combination of hyperbola and parabola. For example, if we look at the graph of the equation below (the first type of equations above),

$\frac{y^2}{9}-x^2=\frac{z}{2}\;\Leftrightarrow\; x^2-\frac{y^2}{9}=\frac{z}{-2}$
and

'slice' the surface horizontally using planes $z=\pm p$ , we get hyperbola shapes (the leftmost graph below). Take note at z = 0, we get a cross, 'x'. z = 0 serves as a threshold whether the direction of hyperbola opening changes its direction.
'slice' the surface vertically using planes $y=\pm q$ , we get parabola shapes (the middle graph below);
'slice' the surface vertically using plane x = 0, we get a parabola (the rightmost graph below). Even if you 'slice' the surface vertically with $x=\pm r$ , you will still get parabolas. To avoid the graph looks too 'crowded', I only use one plane to 'slice' the surface.

Hyperbolic paraboloid is also called as saddle due to its shape. Referrng to this example, imagine that there is an invisible horse facing the y-axis direction with its saddle on it, with the sitting place facing up (z > 0).

You may change the values of a, b and c to see the changes of the graph. Type 1 is the first equation stated on top of this article, Type 2 is the second and Type 3 refers to the third equation. Please un-tick all choices before you change to another type of equation. You may click here if you want to have better viewing experience.

If the surface is shifted to be centered at (h, k, l), then the equations above will look like the equations below:

$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=\frac{z-l}{c}$
$\frac{(x-h)^2}{a^2}-\frac{(z-l)^2}{c^2}=\frac{y-k}{b}$
$\frac{(y-k)^2}{b^2}-\frac{(z-l)^2}{c^2}=\frac{x-h}{a}$

You may refer to other quadric surfaces on the surface shifting concept. It applies the same here.

Betty, created with GeoGebra and Thomas Calculus, Pearson.

Thursday, May 14, 2020

Elliptical Paraboloid

An elliptical paraboloid is a type of quadric surfaces. It has an elliptical opening. Somehow, the opening can be circular sometimes, depending on the values of a, b and c. These values will also affects the direction of the opening, either towards the positive side of the axis or the other way round. If its vertex falls on the origin, then its equation could be one of the equations below.

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z}{c}$ $\frac{x^2}{a^2}+\frac{z^2}{c^2}=\frac{y}{b}$ $\frac{y^2}{b^2}+\frac{z^2}{c^2}=\frac{x}{a}$

If the vextex falls at (h, k, l), then its equation could be one of the equations below:

$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=\frac{z-l}{c}$
$\frac{(x-h)^2}{a^2}+\frac{(z-l)^2}{c^2}=\frac{y-k}{b}$
$\frac{(y-k)^2}{b^2}+\frac{(z-l)^2}{c^2}=\frac{x-h}{a}$

The box below contains all the graphs mentioned above. Tick one category at a time to view the graph. You may slide the values of a, b, c, h, k and l to see the changes on the graph. Observe the following matters:

What is the axis of the graph?
What is the opening of the graph? Elliptical or circular? How do the values of a, b and c affect the opening.
What are the intercepts?
What's the effect of a, b and c when it's positive or negative?

If you wish to have a better view and want to rotate the graph, just click here.

Betty, created with GeoGebra and Thomas Calculus, Pearson.

Pages

Sunday, May 31, 2020

Friday, May 29, 2020

Wednesday, May 27, 2020

Sunday, May 24, 2020

Saturday, May 23, 2020

Friday, May 22, 2020

Wednesday, May 20, 2020

Tuesday, May 19, 2020

Friday, May 15, 2020

Thursday, May 14, 2020