In trying to get an intuition for curl and divergence, I've understood that in the case of R2, div f(x,y) = 2Re( d/dz f(z,z_)) and curl f(x,y) = 2Im( d/dz f(z,z_)), where f(z,z_) is just f(x,y) expressed in z and z conjugate (z_). Is there any way of proving the fundamental properties of div and...
In trying to get an intuition for curl and divergence, I've understood that in the case of R2, div f(x,y) = 2Re( d/dz f(z,z_)) and curl f(x,y) = 2Im( d/dz f(z,z)), where f(z,z) is just f(x,y) expressed in z and z conjugate (z). Is there any way of proving the fundamental properties of div and...
I need the math tools to understand and analyze sequences and their convergence. I know for example that the fibonacci series can be rewritten such that we can calculate for example nr 153 without knowledge of previous numbers. What math subjects is needed to take care of more complicated...
Im talking about all these identities, is there a branch of mathematics that simplifies the proofs of these, and let's me avoid expending the vectors and del operators?
Is there a way to simplify the proof of different vecot calculus identities, such as grad of f*g, which is expandable. And also curl of the curl of a field. Is there a more convenient way to go about proving these relations than to go through the long calculations of actually performing the curl...
Im trying to understand helmholts decomposition, and in order to do so, I feel the need to understand the different ways to apply the del operator to a vector valued function. The dot product and the cross product between two ordinary vectors are easy to understand, thinking about them as a...
Is there a way to motivate the sinus and cosinus functions by looking at their Taylor expansion? Or equivalently, is there a way to see that complex numbers adds their angles when multiplied without knowledge of sin and cos?
Dr Courtney, I understand but your statement implies that we know that e-i2π = 1, and from the definition of eix by its taylor expansion, how can we see on its derivatives that its going to be periodic and perhaps also how do we see from this definition that it describes a circle? The same...
I want to have a simple and intuitive explanation of why the sin and cos waves have such a simple repetitive values for their derivatives at a specific point. Their derivative values are also periodic in respect to the derivative order. For example, e^-x is also periodic, but its derivatives are...
Yes, I know about that condition, but how does that imply that the derivate value in one point is independent of the direction in which it is approached? What does being continuously differentiable have to do with that?
I was told an analytic complex functions has the same derivation value at z0 (random point) however you approach z0. The cauchy riemann eq. shows that z0 has the same derivate value from 2 directions, perpendicular to each other. However, at least some real functions can have the same derivate...
Thanx, but I meant to integrate the legendre with another function, <f,P(n)>, the inner product. My function is x^2, so the integral/inner product will be <x^2,P(n)>, to find the coefficients for a series-expression of x^2 with legendre as the base.