What's the hardest Calculus course?

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21

<p>"In general, a real-valued function of n variables is a rule f that assigns a real number f(x1, x2,..., xn) to each ordered n-tuple (x1, x2,..., xn) in a subset D of n-dimensional space."</p>

<ul>
<li> Calculus:One and several variables, Salas, Hille, and Etgen pg 821</li>
</ul>

<p>Your example is for radius = 1. The real equation is:</p>

<p>r(x,y,z)^2 = x^2 + y^2 +z^2</p>

<p>When you fix a radius, the problem becomes 2D.</p>

<p>"The term "sphere" refers to the surface only, so the usual sphere is a two-dimensional surface."</p>

<ul>
<li> mathworld.wolfram.com</li>
</ul>

22

<p>"That would be 4D since f is a function of the other variables."</p>

<p>I don't believe this. Although a 5D only holds 3 spacial dimensions, one time dimension, and a fifth parameter that varies with all other 4 dimensions.</p>

<p>It's like saying z = x + y is 2D because z depends on x and y. </p>

<p>Flow fields are 4D. Time parameterized flow fields are 5D.</p>

23

<p>^ agreed with rocketDA.</p>

<p>the two-dimensional surface refers to when you unfold the sphere into a planar surface because there is nothing in between the center and the surface. however, you cannot define a sphere without the use of 3 coordinates. you can just as easily define a "ball" which is a 3D object if you supplement the = sign with an inequality. because a sphere is graphed in R3, it is considered 3 dimensional.</p>

24

<p>The dimensionality of the problem isn't defined by how many dimensions you need to visualize something.</p>

25

<p>right, but a 5D graph would require the use of 5 different parameters being varied for complete visualization. (x, y, z, color, time)</p>

26

<p>I agree that the graph would be 5D but I'm saying the dimensionality of the problem isn't 5D.</p>

<p>Were we talking about different things?</p>

27

<p>Out of Calc I, Calc II, Calc III & Diff Eq....I would say Calc III.</p>

<p>Then again (and I was a math major as an undergrad)...I hate vectors.</p>

<p>The hardest Calculus class period? That would be Advanced Calculus which was almost all proofs. I still remember the prof of that course trying have us prove the A times Zero = Zero. That is a 10-step proof using axioms. I was like "I am paying you to teach me this???"</p>

28

<p>Wouldn't using linear algebra be easier to prove A * 0 = 0 ?</p>

29

<p>Sorry....I shoud have used a lower-case "a"....where "a" is a real number.</p>

30

<p>I know "A" is a real number. :-P</p>

31

<p>Calc II, Differential Equation, Calc III then Calc I</p>

<p>Calc II is the hardest, Calc III is just Calc I with z-axis</p>

32

<p>
[quote]
the two-dimensional surface refers to when you unfold the sphere into a planar surface because there is nothing in between the center and the surface. however, you cannot define a sphere without the use of 3 coordinates. you can just as easily define a "ball" which is a 3D object if you supplement the = sign with an inequality. because a sphere is graphed in R3, it is considered 3 dimensional.

[/quote]
</p>

<p>Not that it really matters, or is even on topic, but no. Spheres are 2-dimensional. The reasoning becomes crystal-clear when looking at it from a linear algebra + differential geometry point of view(which I will not descend into). Similarly, (x, y, z, f(x, y, z), t) is 4-D.</p>

<p>Now, on-topic. PDE's (partial differential equations) is probably the hardest math for me, partly because I have the least experience with the, partially because you have to do lots of practice problems to master them, and finally because they're just plain boring from a conceptual point of view.</p>

<p>Lots of people don't like Calc II and III, but I rather enjoyed calc 3.</p>

33

<p>I hate the one I'm in now (Calc I). They teach the class as if everyone has had calculus before, but I haven't had it. Also, no calculators (though, you don't really NEED them, but they would make a few things easier).</p>

34

<p>"Wouldn't using linear algebra be easier to prove A * 0 = 0 ?"</p>

<p>Well this was a pure theoretical course, so you HAD to use axioms and other theorems.</p>

<p>But I see your point....why go through all that just to prove it.</p>

35

<p>Calc II wasn't bad at all.</p>

<p>Once you start learning the different integration techniques...you are fine.</p>

36

<p>It will probably vary a lot from school to school and teacher to teacher but I'd say for me Calc II was the hardest.</p>

37

<p>"Not that it really matters, or is even on topic, but no. Spheres are 2-dimensional. The reasoning becomes crystal-clear when looking at it from a linear algebra + differential geometry point of view(which I will not descend into). Similarly, (x, y, z, f(x, y, z), t) is 4-D."</p>

<p>I still don't agree with you.</p>

<p><a href="http://infohost.nmt.edu/%7Epharis/Dynamic/Chapter3.pdf#search=%22five%20dimensional%20systems%22%5B/url%5D"&gt;http://infohost.nmt.edu/~pharis/Dynamic/Chapter3.pdf#search=%22five%20dimensional%20systems%22&lt;/a>
<a href="http://en.wikipedia.org/wiki/Fifth_dimension%5B/url%5D"&gt;http://en.wikipedia.org/wiki/Fifth_dimension&lt;/a&gt;&lt;/p>

<p>I guess we'll agree to disagree?</p>

38

<p>You can think of a sphere as being two dimensional if you describe it as a circular planar figure that has been rotated.</p>

39

<p>for some reason I thought that DE was really easy...it's basically</p>

<p>1) remember a bunch of rules
2) perform integral</p>

<p>:rolleyes:</p>

<p>Linear Algebra on the other hand was annoying</p>

40

<p>
[quote]
Calc II wasn't bad at all.</p>

<p>Once you start learning the different integration techniques...you are fine.

[/quote]
Depending on teacher you can get destroyed in Calc II.</p>