Given ϕ = x 2 − y z and the point P ( 3 , 4 , 1 ) , find (a) ∇ ϕ at P ; (b) a unit vector normal to the surface ϕ = 5 at P ; (c) a vector in the direction of most rapid increase of ϕ at P ; (d) the magnitude of the vector in (c); (e) the derivative of ϕ at P in a direction parallel to the line r = i − j + 2 k + ( 6 i − j − 4 k ) t .
Given ϕ = x 2 − y z and the point P ( 3 , 4 , 1 ) , find (a) ∇ ϕ at P ; (b) a unit vector normal to the surface ϕ = 5 at P ; (c) a vector in the direction of most rapid increase of ϕ at P ; (d) the magnitude of the vector in (c); (e) the derivative of ϕ at P in a direction parallel to the line r = i − j + 2 k + ( 6 i − j − 4 k ) t .
Given
ϕ
=
x
2
−
y
z
and the point
P
(
3
,
4
,
1
)
,
find
(a)
∇
ϕ
at
P
;
(b) a unit vector normal to the surface
ϕ
=
5
at
P
;
(c) a vector in the direction of most rapid increase of
ϕ
at
P
;
(d) the magnitude of the vector in (c);
(e) the derivative of
ϕ
at
P
in a direction parallel to the line
r
=
i
−
j
+
2
k
+
(
6
i
−
j
−
4
k
)
t
.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
(a) Find the directional derivative of z = x²y at (5,5) in
the direction of π/2 with the positive x-axis.
(b) In which direction is the directional derivative the
largest at the point (5, 5)? Enter your answer as a vector
whose length is the largest value of the directional
derivative.
Can you help me with this question
Find the directional derivative Duf(1, 1), where f(x, y) = x2 + y2 and u is the unit vector at an angle of π/6 from horizontal.
Fundamentals of Differential Equations and Boundary Value Problems
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