- #1

Rippling Hysteresis

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- Homework Statement
- f(x,y,z) = z^k(e^(x^2+y^2) -1))/(x^2+y^2+z^2)^k for when (x,y,z) does not equal (0,0,0). f(x,y,z)=0 otherwise.

a) Find all the real, positive values of k where the function is continuous at the origin. Thus, find the values of k where the limit at (x,y,z) -> (0,0,0) equals zero.

b) Find the values of k where each of the partials, fx, fy, fz evaluated at (0,0,0) exist. Find the values of these partials

- Relevant Equations
- L'Hopital?

Parameterization?

fx(0,0) = lim t->0 = f(t,0) -f(0,0)/t

(a) I thought perhaps a parameterization would be the place to begin given all the squared terms.

x=rcos(u)sin(v)

y=rsin(u)sin(v)

z=rcos(v)

That would yield: r^k(cos(v))^k*(e^(r^2*(sin(v))^2))/(r^(2k))

Canceling a r^k at each level: (cos(v))^k*(e^(r^2*(sin(v))^2))/(r^(k))

I'm not sure how important the trig terms are, since they are bounded between (-1,1), but they may be necessary to keep. My focus is drawn to the e^(r^2)/r^k term. When I graph that, regardless of exponent, that terms since to tend toward infinity as r->0, which makes sense. So I'm not sure what form of analysis or technique I should use to further simplify this.

Intuition from 1-D Calc says maybe L'Hopital? But a little unsure how to apply with MV, or if it's even relevant. w.r.t. 'r' the top would become (cos(v))^k*(e^(r^2*(sin(v))^2))*(2r*(sin(v))^2) and the bottom would become (k-1)*r^(k-1). I'm not sure how that helps, because then it's still effectively (0)(infinity)/(infinity).

(b) I have a feeling this part would depend on part (a), and I should use the limit definition formula.

x=rcos(u)sin(v)

y=rsin(u)sin(v)

z=rcos(v)

That would yield: r^k(cos(v))^k*(e^(r^2*(sin(v))^2))/(r^(2k))

Canceling a r^k at each level: (cos(v))^k*(e^(r^2*(sin(v))^2))/(r^(k))

I'm not sure how important the trig terms are, since they are bounded between (-1,1), but they may be necessary to keep. My focus is drawn to the e^(r^2)/r^k term. When I graph that, regardless of exponent, that terms since to tend toward infinity as r->0, which makes sense. So I'm not sure what form of analysis or technique I should use to further simplify this.

Intuition from 1-D Calc says maybe L'Hopital? But a little unsure how to apply with MV, or if it's even relevant. w.r.t. 'r' the top would become (cos(v))^k*(e^(r^2*(sin(v))^2))*(2r*(sin(v))^2) and the bottom would become (k-1)*r^(k-1). I'm not sure how that helps, because then it's still effectively (0)(infinity)/(infinity).

(b) I have a feeling this part would depend on part (a), and I should use the limit definition formula.

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