Find the directional derivative of the function defined by
at
in the direction
.
To find the directional derivative of the function \( f : \mathbb{R}^4 \rightarrow \mathbb{R}^3 \) defined by \( f(x, y, z, w) = (x^2y, xyz, x^2 + y^2 +zw^2) \) at the point \( a = (1,2,-1,-2) \) in the direction \( v = (0,1,2,-2) \), we can use the formula for the directional derivative:
\[ D_vf(a) = \nabla f(a) \cdot v \]
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Which of the following sets are connected sets in 2 R with the metric given against it?
Justify your answer.
i) under the standard metric.
ii) under the discrete metric.
Show that the components of a metric space is either identical or pairwise disjoint.
State whether the following statements are True or False. Justify your answers.
a) The sequence is convergent in
under the discrete metric on
.
b) A subset in a metric space is compact if it is closed.
c) Continuous image of a path connected space is path connected.
d) The second derivative of a linear map from to
never vanishes.
e) If for all
, then
.
Let A and B be any two subsets of a metric space , then show that
i) int {E : is open and
}
ii) int = int A
int B
iii) int(A )B
int A
int B
iv) .
Which of the following sets are totally bounded? Give reasons for your answer. Are they compact?
i) 2N in (N,d) where d is the discrete metric.
ii) in (R,d) where d is the Euclidean metric.
Near what points may the surface be represented uniquely as a graph of a differentiable function
? Locate such a point.
Use the method of Lagrange’s multiplier method to find the shortest possible distance from the ellipse to the line
.