Your solid angle here is essentially a rectangular pyramid with the tip at the origin. The base is $\sin \theta d\theta \times d\phi$ and you want the normal to be within that pyramid. It is no different from the cone case except for the shape of the region.
A dihedral angle is the angle between two intersecting planes. In chemistry it is the angle between planes through two sets of three atoms, having two atoms in common. In solid geometry it is defined as the union of a line and two half-planes that have this line as a common edge. In higher dimension, a dihedral angle represents the angle between two hyperplanes.
== Definitions ==
A dihedral angle is an angle between two intersecting planes on a third plane perpendicular to the line of intersection.
A torsion angle is a particular example of a dihedral angle, used in stereochemistry to define the...
Problem
If we have surfaces $$ z=8-x^3-y^2 $$ $$z= \frac{23}{8}+\frac{y}{16}-\frac{\sin(\pi x)}{4\pi} $$
Show that these surfaces intersect in point $(1,2,3)$ and that they are perpendicular at this point.
Attempt to solve
I know that if two planes are perpendicular their normal vector's dot...
In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable.
Bayes' theorem calculates the renormalized pointwise product of...
In Bayesian statistics, the posterior probability of a random event or an uncertain proposition is the conditional probability that is assigned after the relevant evidence or background is taken into account. Similarly, the posterior probability distribution is the probability distribution of an unknown quantity, treated as a random variable, conditional on the evidence obtained from an experiment or survey. "Posterior", in this context, means after taking into account the relevant evidence related to the particular case being examined.
== Definition ==
The posterior probability is the probability...
In the theory of partial differential equations, an a priori estimate (also called an apriori estimate or a priori bound) is an estimate for the size of a solution or its derivatives of a partial differential equation. A priori is Latin for "from before" and refers to the fact that the estimate for the solution is derived before the solution is known to exist. One reason for their importance is that if one can prove an a priori estimate for solutions of a differential equation, then it is often possible to prove that solutions exist using the continuity method or a fixed point theorem.
A priori...
1
