Find the directional derivation of the function defined by
at in the direction~
.
The directional derivative of a differentiable function f: R^n → R in the direction of a unit vector u = (u1, u2, ..., un) at a point x = (x1, x2, ..., xn) is given by:
D_uf(x) = ∇f(x) · u
where ∇f(x) is the gradient of f at x.
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Define components in a metric space. What are all the components of the set of all nonzero real numbers under the
1) usual metric on R, and
2) the discrete metric on R?
The functiongiven by
is differentiable at 0.
Define and give an example for each of the following concepts in the context of signals and systems:
i) A stable system
ii) A time-varying system
Define saddle points. Compute the saddle points of the function given by
.
The function is not uniformly continuous.
Let f,g ∈L′(R), define convolution f *g of f and g. Show that if either f or g is bounded, then the convolution f *g exists for all x in R and is bounded in R .
If f: X → Y is a continuous map between metric spaces X and Y and K is a compact subset of X, then show that f(K) is compact.