From the toolbar do the following: ``format'' followed by ``style'', the dropdown menu offers you lots of options. ``input'' is the default, and is the ``format'' in which you do calculations. However Mathematica also allows various fonts and styles of text input. In this problem set you should include a text ``cell'' before each problem which identifies the problem (e.g. Problem 1), and a text cell at the top of the page with your name and the number of the worksheet (e.g. ``worksheet 2'').
Last week we did derivatives and integrals using the
full Mathematica commands. However many of these commands
may be entered from ``palettes''. To activate a palette,
from the toolbar do the following: ``file'' followed by
``palettes''. You have several options: ``basic input'' is
one I like.
The reference material for this worksheet is sections 1.2.1 - 1.2.6 of the practical introduction to mathematica. Your should read this material as you work through the excersizes below.
Lists and Vectors
By now you must have read about vectors. A vector is a quantity
which, unlike a scalar, can have many components. For
example in Newton's second law of motion
In Mathematica vectors are represented in the same way. In Mathematica this object is called a list, because it can be used for more general objects such as matrices and tensors. In this worksheet we just work with vectors. Type ``A = Ax, Ay, Az''. This means Mathematica associates the object A with the list Ax, Ay, Az. Now type ``B = Bx, By, Bz''. Type ``Dot[A,B]''. This will give the dot product =AxBx+AyBy+AzBz which is the same as , where is the angle between the vectors and .
Likewise, the cross product of two vectors ( ) yields another vector AyBz-AzBy, AzBx-AxBz, AxBy-AyBx. ``Type Cross[A,B]'' and verify that you indeed get the above expression in terms of the components of and . Unit vectors can be easily written with lists as: 1,0,0, 0,1,0, 0,0,1. Check with Mathematica that .
You can see that the elements in the list Ax, Ay, Az of the vector are its x, y, and z components. How do we access the individual components from A?.
Type ``A[]'' and check that this gives Ay. How would you get Mathematica to print out the second element of the cross product ``Cross[A,B]''?
Assignment 2. - Hand in by Thursday Jan. 27
Problem 1. Consider two vectors , and . Using Mathematica:
(i) Check that they are both of unit magnitude.
(ii) Find .
(iii) Find the angle between these two vectors.
(iv) Find the cross pruduct of these two vectors.
Problem 2. Consider the unit vectors along x, y, and z directions: 1,0,0 0,1,0 0,0,1. Verify: , , .
Problem 3. Verify that for any three vectors , , and that .
Problem 4. The motion of a particle is given by =a(cos(t) + sin(t)) Find its velocity . Calculate , where , and and verify . Do you recognise this motion? Plot the motion to confirm your intuition (use the help menu to look up how to use the command ``ParametricPlot'' for this problem - you will need to choose values for a and .).