A continuous function is a function whose value of function at a point is equals to the value of limit at that same point. We can write the condition of continuity as: {eq}\displaystyle { g(b) = \lim_{x \to b} g(x) } {/eq} ...
How to prove a manifold is differentiable?Differentiable Manifolds:A manifold {eq}M^n {/eq} is usually understood as a topological manifold, meaning that {eq}M^n {/eq} is a separable topological space together with an atlas of homeomorphisms (charts) {eq}\phi_x:U_x\to \mathbb{R}^n...
How to prove that if a relation r is transitive, then it is also quasi transitive? Does the relation y(x + y) = 1 specify a function? Explain. Does the relation x(x+y)=1 specify a function? Explain. Prove or give a counterexample: If R ...
FAQ: How Do You Prove the Multivariable Chain Rule? 1. What is the multivariable chain rule? The multivariable chain rule is a formula used to compute the derivative of a composite function when the function depends on multiple variables. It allows us to differentiate functions that are ...
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In summary, the conversation discusses how to prove a rule for differentiable curves in $\mathbb{R}^3$ involving cross products. Two different proofs are given and it is determined that to use the product rule, the cross product must be written as a determinant. The correctness of one of ...
Typically, a differentiable nonlinear activation function is used in the hidden layers of a neural network. This allows the model to learn more complex functions than a network trained using a linear activation function. In order to get access to a much richer hypothesis space that would benefit...
For concave functions, the hypograph (the set of points lying on or below its graph) is a closed set. Closed Function Examples As well as convex functions, continuous on a closed domain, there are many other functions that have closed set epigraphs. For example, all differentiable convex fu...
how to prove a function is strictly decreasing Decreasing Functions:The one-one function is either increasing or decreasing. But one-one onto function is strictly increasing or strictly decreasing function. The function is tested with the first derivative for this kind of behavior test....
How to prove that a function to be analytic?Analytic FunctionsLet f(z) is a complex valued function defined in a domain D then the function f(z) is said to be analytic if it is differentiable everywhere in the complex plane C. Analytic functions must be holomorphic....