Solve Schaum's Vector Analysis Prob 4.65: Find Constants a,b,c

AI Thread Summary
To solve Schaum's vector analysis problem 4.65, the goal is to find constants a, b, and c such that the directional derivative of the function phi=axy^2+byz+cz^2x^3 at the point (1,2,-1) has a maximum magnitude of 64 in the z-axis direction. The gradient of phi is calculated as (4a+3c)i+(4a-b)j+(2b-2c)k, indicating that for maximum directionality along the z-axis, the x and y components must be zero. This leads to the equations (2b - 2c) = 64 and (4a+3c) = (4a-b) = 0. Solving these equations provides the values for a, b, and c. The solution confirms the conditions for maximum directional derivative are satisfied.
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Homework Statement


Schaum's vector analysis prob 4.65
Find the values of constants a,b,c so that directional derivative of phi=axy^2+byz+cz^2x^3 at (1,2,-1) has a maximum of magnitude 64 in a direction parallel to the z axis.


Homework Equations





The Attempt at a Solution


Directional derivative is maximum along grad(phi). After taking grad(phi) at (1,2,-1) I got
(4a+3c)i+(4a-b)j+(2b-2c)k
I don't know how to make use of the 'direction parallel to z axis' condition.
 
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If grad(phi) is maximum in Z-axis, it may be zero in x and y-axis. Then
(2b - 2c) = 64.
(4a+3c) = (4a-b) = 0.
Solve these equations to find a, b and c.
 
Got it! Thanks.
 
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