Learn English – Should there be capitalization after an equation

This can go either way. If you are starting a complete sentence that represents a summation of what came before, you are certainly entitled to capitalize the sentence.

Here's an example of such a sentence: There is a sentence after the preceding colon.

But if you are using the colon to offset a list or other material that does not form a complete sentence, it makes no sense to capitalize what follows:

Here are some examples of breakfast foods: eggs, bacon, toast, hash browns, orange juice, coffee.

A parenthetical statement that is a complete sentence should begin with a capital. Your examples are wrong, however, because you should begin by capitalizing the first letter of the abbreviation i.e. or e.g.:

Here is my technical sentence. (E.g., should the word of this example be capitalized like this?)

Here is another technical sentence. (I.e., this really isn't an i.e., I just want to know if the first word should or should not be capitalized and am providing an example of yes and no.)