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

In summary, the problem asks to find the values of constants a, b, and c that will result in a maximum directional derivative of magnitude 64 at the point (1,2,-1) in the direction parallel to the z axis. After taking the gradient of the given function, the conditions for maximum gradient in the z direction are (2b-2c) = 64 and (4a+3c) = (4a-b) = 0. Solving these equations will give the values of a, b, and c.
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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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  • #2
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.
 
  • #3
Got it! Thanks.
 

1. How do I approach solving Schaum's Vector Analysis Prob 4.65?

To solve Schaum's Vector Analysis Prob 4.65, you can follow these steps:

  • Start by understanding the problem and the given information.
  • Identify what is being asked in the problem.
  • Use vector analysis principles and formulas to manipulate the given information and find a solution.
  • Check your answer by plugging it back into the original problem.

2. What are the constants a,b,c and how do I find them?

The constants a, b, and c represent the coefficients of the unknown variables in the problem. To find them, you can use the given information and solve for these variables using algebraic manipulation.

3. Is there a specific method or formula to use for solving this problem?

Yes, for Schaum's Vector Analysis Prob 4.65, you can use the formula for finding the angle between two vectors in 3D space, as well as the formula for finding the magnitude of a vector. These can be found in most vector analysis textbooks or online resources.

4. Can I use a calculator to solve this problem?

Yes, you can use a calculator to solve this problem. However, it is important to understand the principles and formulas behind vector analysis and not solely rely on the calculator to solve the problem.

5. Are there any common mistakes to avoid when solving this type of problem?

Yes, some common mistakes to avoid when solving Schaum's Vector Analysis Prob 4.65 are:

  • Forgetting to convert angles from degrees to radians.
  • Not using the correct formula for finding the angle between two vectors.
  • Accidentally using the magnitude of one vector instead of the other.
  • Not double-checking the final answer with the original problem.

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