Mian Systems LLC
Directional Derivatives (Contd.)
Solve Example: Suppose a rectangular coordinate system is located in space such that the temperature T at the point
(x,y,z,) is given by the formula T= 100/(x2+y2+z2).
Find the rate of change of T w.r.t. distance at the point P(1,3,-2) in the direction of the vector a = i –j +k.
In what direction from P does T increases most rapidly? What is the maximum rate of change of T at P?
Solution:
Step 1- Find magnitude. It is same as the Pythagorean Theorem.
Step 2- Find Unit vector => u= I/||a||*a
Step 3- partial derivative
Step 4- Du(f,x)
Step 1 √12+12+12 = √3
Step 2 U = 1/√3(i – j + k)
Step 3 Partial derivative
(x,y,z,) is given by the formula T= 100/(x2+y2+z2).
Find the rate of change of T w.r.t. distance at the point P(1,3,-2) in the direction of the vector a = i –j +k.
In what direction from P does T increases most rapidly? What is the maximum rate of change of T at P?
Solution:
Step 1- Find magnitude. It is same as the Pythagorean Theorem.
Step 2- Find Unit vector => u= I/||a||*a
Step 3- partial derivative
Step 4- Du(f,x)
Step 1 √12+12+12 = √3
Step 2 U = 1/√3(i – j + k)
Step 3 Partial derivative
Step 4 Apply theorem
Duf(x,y,z) = ∇f(x,y,z).u
Which is the same as
Du f(x,y,z) = fx(x,y,z)u1 +fy(x,y,z)u2+ fz(x,y,z)u3 OR Du f(x,y,z) = fx(x,y,z)a +fy(x,y,z)b+ fz(x,y,z)c
OR Du f(x,y,z) = fx(x,y,z)i +fy(x,y,z)j+ fz(x,y,z)k
Duf(x,y,z) = ∇f(x,y,z).u
Which is the same as
Du f(x,y,z) = fx(x,y,z)u1 +fy(x,y,z)u2+ fz(x,y,z)u3 OR Du f(x,y,z) = fx(x,y,z)a +fy(x,y,z)b+ fz(x,y,z)c
OR Du f(x,y,z) = fx(x,y,z)i +fy(x,y,z)j+ fz(x,y,z)k
will escape from the region at P. The boundary us then said to be insulated at the point P. The region is insulated
along the part of a boundary if it is insulated at every point on that part. Analogous statements can be made for
three dimensional regions. END!
Solve Ex. f(x,y) = 3x-5xy+10y; P(2,1) and vector <-2,1>
Solve Ex. f(x,y) = x2+2xy+3y2; P(2,1) and vector <1,1>
Solve EX. f(x,y) = x2+xy+y2; P(-1,1)
Find the direction of maximum increase or decrease using the gradient vector
Source:http://www.youtube.com/watch?v=XQNISK-0PQw
(There is a mistake in the video in the last step when he is negating the vector.)
Solve EX. F(x,y,z) = x2 + y2 + z2
a) Sketch level surface
b) Sketch slope at various points
Source: http://www.youtube.com/watch?v=Li05zuaI7C4
along the part of a boundary if it is insulated at every point on that part. Analogous statements can be made for
three dimensional regions. END!
Solve Ex. f(x,y) = 3x-5xy+10y; P(2,1) and vector <-2,1>
Solve Ex. f(x,y) = x2+2xy+3y2; P(2,1) and vector <1,1>
Solve EX. f(x,y) = x2+xy+y2; P(-1,1)
Find the direction of maximum increase or decrease using the gradient vector
Source:http://www.youtube.com/watch?v=XQNISK-0PQw
(There is a mistake in the video in the last step when he is negating the vector.)
Solve EX. F(x,y,z) = x2 + y2 + z2
a) Sketch level surface
b) Sketch slope at various points
Source: http://www.youtube.com/watch?v=Li05zuaI7C4
