7. If vector [tex]f(x y z)=x y i+y z j+z x k[/tex] what is the directional derivative of vector [tex]f[/tex] at point [tex](1,1,1)[/tex] in the direction of vector [tex]( i + j + k )[/tex]
(a) [tex]2 \sqrt{3}[/tex]
(b) [tex]2 \sqrt{2}[/tex]
(c) [tex]3 \sqrt{2}[/tex]
(d) [tex]3 \sqrt{3}[/tex]
Answer (1)
Calculate the gradient of the vector field f ( x , y , z ) = x y i + yz j + z x k , which is ∇ f = y i + z j + x k .
Evaluate the gradient at the point ( 1 , 1 , 1 ) , resulting in ∇ f ( 1 , 1 , 1 ) = i + j + k .
Find the unit vector in the direction of v = i + j + k , which is u = 3 1 i + 3 1 j + 3 1 k .
Compute the directional derivative as the dot product of ∇ f ( 1 , 1 , 1 ) and u , yielding 3 .
Explanation
Problem Setup We are given the vector field f ( x , y , z ) = x y i + yz j + z x k and asked to find the directional derivative at the point ( 1 , 1 , 1 ) in the direction of the vector v = i + j + k . The directional derivative measures the rate of change of the vector field in the specified direction.
Calculating the Gradient First, we need to find the gradient of the vector field f . The gradient is given by ∇ f = ( ∂ x ∂ ( x y ) , ∂ y ∂ ( yz ) , ∂ z ∂ ( z x ) ) = ( y , z , x ) So, ∇ f = y i + z j + x k .
Evaluating the Gradient at the Point Next, we evaluate the gradient at the point ( 1 , 1 , 1 ) :
∇ f ( 1 , 1 , 1 ) = ( 1 , 1 , 1 ) = i + j + k .
Finding the Unit Vector Now, we need to find the unit vector in the direction of v = i + j + k . The magnitude of v is ∥ v ∥ = 1 2 + 1 2 + 1 2 = 3 So, the unit vector is u = ∥ v ∥ v = 3 1 i + 3 1 j + 3 1 k .
Calculating the Directional Derivative Finally, we compute the directional derivative as the dot product of the gradient at the point and the unit vector: D u f ( 1 , 1 , 1 ) = ∇ f ( 1 , 1 , 1 ) ⋅ u = ( 1 , 1 , 1 ) ⋅ ( 3 1 , 3 1 , 3 1 ) = 3 1 + 3 1 + 3 1 = 3 3 = 3 Therefore, the directional derivative of f at ( 1 , 1 , 1 ) in the direction of i + j + k is 3 .
Final Answer The directional derivative of the vector field f ( x , y , z ) = x y i + yz j + z x k at the point ( 1 , 1 , 1 ) in the direction of the vector i + j + k is 3 .
Examples
Directional derivatives are used in various fields such as physics and engineering. For instance, in fluid dynamics, the directional derivative can determine how the velocity of a fluid changes in a specific direction. Imagine you're analyzing the flow of air around an airplane wing; the directional derivative helps you understand how the wind speed varies at different points and directions, which is crucial for optimizing the wing's design and improving aerodynamic performance. This concept allows engineers to make informed decisions about the shape and orientation of structures in fluid environments.
