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If three vectors in R3 are linearly independent then they form a basis for R3; they span R3.

I feel this thread should not end without it being pointed out that a forward-slash, "/", rather than a backslash, "\", may be used to indicate division.

Re: Help Finding Limits!!! G'day, Roksteady. For future posts, be sure to use parentheses. In this case the expression in the limit is typed: (3e^(3x) + 2e^x - 1)/(e^(3x) - e^(2x) + 5). To find the limit, simply divide both the numerator and the denominator by e^(3x), and apply the "limit laws".

If r(t) is the position vector, then the unit tangent vector is \frac{r'(t)}{||r'(t)||}.

Your equation labelled (3) and the equation that you obtained from (1) and (2) are conditions involving \alpha, \beta, \gamma, \delta, x* and y* (replace x and y), but not \lambda. Isn't that all that is asked of you?

Now apply the definition of * to (x+y-xy)*z, Jessica. Remember that you can also expand the right-hand side of (x *y)*z= x *(y *z) so you know where you're going.

Hi Tango, Recall the identity: x^3 \pm y^3 = (x \pm y)(x^2 \mp xy + y^2). An acronym due to Soroban helps us to remember the signs in the right-hand side: S.O.A.P. for Same-Opposite-Always Positive. So, returning to the initial expression, 8 - (a + 1)^3, your first task is to choose...

Hi Anatolyz, To begin to write an answer to the question "1) Prove that if A^2=0, then the columns of the matrix A are vectors in the solution space of the system Ax=0", begin by writing something like: "Let v be a column of A." Your task is then to show that Av = 0. To do so, remember that we...

Hi ImOK, If you are ok with their derivation of the identity for sin(5x), the algebraic approach at http://mathworld.wolfram.com/TrigonometryAnglesPi5.html may be of use to you.

Hi Green_tea, Since axb is orthogonal to a and b, we must have that axb = kb'' where k is some scalar. Furthermore, from your diagram it appears that b'' is chosen so that k is non-negative. (*) This implies that |a x b| = k|b''|. But |b''|= |b|sin(v) so we have |a x b| = k|b|sin(v). (**) We...

Ignore Stoke's theorem for the moment. (1) and (3) are the same kind of expression; the former is just parameterised. And (2) and (4), where the dS is a vector, are the same by the definition of a surface integral, which I am sure you will find in your notes.

(1) To define continuity one needs a topology to work with. (If you are not familiar with the term "topology", replace it with "metric"). In this context the usual thing to do is to consider an nxn matrix as an n^2-tuple (a vector with n^2 entries) and to use the Euclidean metric. The author...

It is a basic fact that a bijection from a set A to a set B has an inverse that is a bijection from B to A.

The final answer of 0 is correct. I meant to say the Divergence theorem (not Stokes', which involves the curl); using that agrees with your final answer.

Re: Surface integral If you are familiar with the divergence theorem, that's slightly neater. Otherwise, to calculate the surface integral directly, then yes, spherical coordinates are the way to go to parameterise the surface. Let's see how you go; show your work if you get stuck. Edit...

Hi Dts, Your work so far is good. Notice the transformation you have chosen is a clockwise rotation by pi/4. Call the originial integral I. Then, after substituting for x and y, using the symmetry of the rotated square and integrating first with respect to v, we have I =...

I assume Q are the rational numbers as in your post http://www.physicsforums.com/showthread.php?t=273566 at Physics Forums. Since 0 = 0 + 0\sqrt{5}, and 0\in Q, we have that 0 belongs to Q(\sqrt{5}).

This step isn't right. You should get x_1'' - 9x_1 = 0.

From (*), you can substitute x_2 = 1/2*x_1' - 1/2*x_1 into (#).

If h is a function of t, then h(2) should be a number; e.g., if h(t) = t^2, then h(2) = 4. So h(2) = 5cos(t) doesn't make sense; are you sure you have this written correctly? My guess is it would just be h(t)=5cos(t). And one minor point: 16*2=32, not 64.