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That is, the Jacobian of a composite function is the product of the Jacobians of the composed functions (evaluated at the appropriate points). The higher-dimensional chain rule is a generalization of the one-dimensional chain rule. If k, m, and n are 1, so that f : R → R and g : R → R, then the Jacobian matrices of f and g are 1 × 1.
In probability theory, the chain rule [1] (also called the general product rule [2] [3]) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities.
Reverse accumulation traverses the chain rule from outside to inside, or in the case of the computational graph in Figure 3, from top to bottom. The example function is scalar-valued, and thus there is only one seed for the derivative computation, and only one sweep of the computational graph is needed to calculate the (two-component) gradient.
The partial derivative of f with respect to x does not give the true rate of change of f with respect to changing x because changing x necessarily changes y. However, the chain rule for the total derivative takes such dependencies into account. Write () = (, ()). Then, the chain rule says
Chain rule Suppose that f : A → R is a real-valued function defined on a subset A of R n, and that f is differentiable at a point a. There are two forms of the chain rule applying to the gradient. First, suppose that the function g is a parametric curve; that is, a function g : I → R n maps a subset I ⊂ R into R n.
This can be derived using the chain rule for derivatives: = and dividing both sides by to give the equation above. In general all of these derivatives — dy / dt , dx / dt , and dy / dx — are themselves functions of t and so can be written more explicitly as, for example, d y d x ( t ) {\displaystyle {\frac {dy}{dx}}(t)} .
To elucidate the connection with the triple product rule, consider the point p 1 at time t and its corresponding point (with the same height) p̄ 1 at t+Δt. Define p 2 as the point at time t whose x-coordinate matches that of p̄ 1, and define p̄ 2 to be the corresponding point of p 2 as shown in the figure on the right.
The Laplacian is a measure of how much a function is changing over a small sphere centered at the point. When the Laplacian is equal to 0, the function is called a harmonic function . That is, Δ f = 0. {\displaystyle \Delta f=0.}