Recent content by Agent 47

Sorry for the vagueness. Here is what I'm referring to: The way it's presented in my textbook makes it seem like the addition of two tangent lines lying on the same plane. What I'm wondering is how adding two lines lying in the same plane actually gives you that plane.

Then my followup question would be how does the addition of the two tangent lines actually equate to the correct tangent plane. Is there a proof for tangent plane approximation that doesn't use the chain rule?

##dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy## I'm confused as to how the total derivative represents the total change in a function. My own interpretation, which I know is incorrect, is that ##\frac{\partial z}{\partial x} dx## represents change in the x...

Okay I appreciate all your help. I get it now. Out of curiosity would it be possible to solve this integral by parameterizing it? I just tried and I know I have to get it into the form ##\int_C f(r(t)) \cdot r'(t)dt## So to get a circle of radius r ##r(t) = r<cost,sint>## ##r'(t) =...

So I guess this leads into another question for me. When do you interpret ##dr## as just ##dr## and when do you interpret it as ##r'(t)##? Does it depend on whether or not I decide to parameterize it?

I think I understand. So is this the integral I was looking for? ##B \int_0^{2 \pi r} dr##

I guess I'm just looking for the intermediate step in a more detailed manner. How do you determine ##\int B \cdot dr = B 2 \pi r##? What are the limits of integration? Are there any limits of integration? Is B interpreted as a constant? I know the beginning of the problem. I know the end. I just...

Homework Statement Experiments show that a steady current I in a long wire produces a magnetic field B that is tangent to any circle in the plane perpendicular to the wire and whose center is the axis of the wire. Ampere's Law relates the electric current to its magnetic effects and states...

Thank you so much. This has been bothering me for a while. So basically I interpreted this as ##f(u,v)## instead of ##f(u)## and ##g(v)## separately. Right?

Homework Statement Show that any function of the form ##z = f(x + at) + g(x - at)## is a solution to the wave equation ##\frac {\partial^2 z} {\partial t^2} = a^2 \frac {\partial^2 z} {\partial x^2}## [Hint: Let u = x + at, v = x - at] 2. The attempt at a solution My problem with this is...