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A.Find the directional derivative of f(x,y,z) = xy² + 4xyz + z² at (1,2,3) in the direction of 3i + 4j -5 overline k 12 3 4 6 b. Find the constants a, b, and c such that overline v =(3x+ay+z) overline i +(2x-y+bz) overline j + (x + cy + z) * kis irrotational?
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A.Find the directional derivative of f(x,y,z) = xy² + 4xyz + z² at (1,...
Directional Derivative
- Given function: f(x,y,z) = xy² + 4xyz + z²
- Point of interest: (1,2,3)
- Direction vector: 3i + 4j - 5k
Now, we need to find the gradient of f at the given point and then take the dot product with the direction vector to find the directional derivative.
- Gradient of f: ∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k
- ∂f/∂x = y² + 4yz, ∂f/∂y = 2xy + 4xz, ∂f/∂z = 4xy + 2z
- ∇f = (2*2 + 4*2*3)i + (2*1 + 4*1*3)j + (4*1*2 + 2*3)k
- ∇f = 20i + 14j + 14k
Now, the directional derivative is given by the dot product of the gradient and the direction vector:
- D_vf = ∇f ⋅ v/|v|
- v = 3i + 4j - 5k
- |v| = √(3² + 4² + (-5)²) = √50
- D_vf = (20*3 + 14*4 - 14*5)/ √50
- D_vf = (60 + 56 - 70)/ √50
- D_vf = 46/ √50
Irrotational Vector Field
To find the constants a, b, and c for the vector field to be irrotational, we need to check if the curl of the vector field is zero. The curl of a vector field F = P i + Q j + R k is given by:
- curl(F) = (∂R/∂y - ∂Q/∂z)i + (∂P/∂z - ∂R/∂x)j + (∂Q/∂x - ∂P/∂y)k
For the given vector field V = (3x+ay+z) i + (2x-y+bz) j + (x + cy + z) k:
- P = 3x + ay + z, Q = 2x - y + bz, R = x + cy + z
- ∂R/∂y = c, ∂Q/∂z = b, ∂P/∂z = 1, ∂R/∂x = 1, ∂Q/∂x = 2, ∂P/∂y = a
For the vector field to be irrotational, the curl should be zero:
- curl(V) = (c - b)i + (1 - 2)j + (2 - a)k
- For curl(V) to be zero, c = b, a = 2
Therefore, the constants a = 2, b = c are such that the vector field is irrotational.
- Given function: f(x,y,z) = xy² + 4xyz + z²
- Point of interest: (1,2,3)
- Direction vector: 3i + 4j - 5k
Now, we need to find the gradient of f at the given point and then take the dot product with the direction vector to find the directional derivative.
- Gradient of f: ∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k
- ∂f/∂x = y² + 4yz, ∂f/∂y = 2xy + 4xz, ∂f/∂z = 4xy + 2z
- ∇f = (2*2 + 4*2*3)i + (2*1 + 4*1*3)j + (4*1*2 + 2*3)k
- ∇f = 20i + 14j + 14k
Now, the directional derivative is given by the dot product of the gradient and the direction vector:
- D_vf = ∇f ⋅ v/|v|
- v = 3i + 4j - 5k
- |v| = √(3² + 4² + (-5)²) = √50
- D_vf = (20*3 + 14*4 - 14*5)/ √50
- D_vf = (60 + 56 - 70)/ √50
- D_vf = 46/ √50
Irrotational Vector Field
To find the constants a, b, and c for the vector field to be irrotational, we need to check if the curl of the vector field is zero. The curl of a vector field F = P i + Q j + R k is given by:
- curl(F) = (∂R/∂y - ∂Q/∂z)i + (∂P/∂z - ∂R/∂x)j + (∂Q/∂x - ∂P/∂y)k
For the given vector field V = (3x+ay+z) i + (2x-y+bz) j + (x + cy + z) k:
- P = 3x + ay + z, Q = 2x - y + bz, R = x + cy + z
- ∂R/∂y = c, ∂Q/∂z = b, ∂P/∂z = 1, ∂R/∂x = 1, ∂Q/∂x = 2, ∂P/∂y = a
For the vector field to be irrotational, the curl should be zero:
- curl(V) = (c - b)i + (1 - 2)j + (2 - a)k
- For curl(V) to be zero, c = b, a = 2
Therefore, the constants a = 2, b = c are such that the vector field is irrotational.
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A.Find the directional derivative of f(x,y,z) = xy² + 4xyz + z² at (1,2,3) in the direction of 3i + 4j -5 overline k 12 3 4 6 b. Find the constants a, b, and c such that overline v =(3x+ay+z) overline i +(2x-y+bz) overline j + (x + cy + z) * kis irrotational?
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A.Find the directional derivative of f(x,y,z) = xy² + 4xyz + z² at (1,2,3) in the direction of 3i + 4j -5 overline k 12 3 4 6 b. Find the constants a, b, and c such that overline v =(3x+ay+z) overline i +(2x-y+bz) overline j + (x + cy + z) * kis irrotational? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A.Find the directional derivative of f(x,y,z) = xy² + 4xyz + z² at (1,2,3) in the direction of 3i + 4j -5 overline k 12 3 4 6 b. Find the constants a, b, and c such that overline v =(3x+ay+z) overline i +(2x-y+bz) overline j + (x + cy + z) * kis irrotational? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A.Find the directional derivative of f(x,y,z) = xy² + 4xyz + z² at (1,2,3) in the direction of 3i + 4j -5 overline k 12 3 4 6 b. Find the constants a, b, and c such that overline v =(3x+ay+z) overline i +(2x-y+bz) overline j + (x + cy + z) * kis irrotational?.
A.Find the directional derivative of f(x,y,z) = xy² + 4xyz + z² at (1,2,3) in the direction of 3i + 4j -5 overline k 12 3 4 6 b. Find the constants a, b, and c such that overline v =(3x+ay+z) overline i +(2x-y+bz) overline j + (x + cy + z) * kis irrotational? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A.Find the directional derivative of f(x,y,z) = xy² + 4xyz + z² at (1,2,3) in the direction of 3i + 4j -5 overline k 12 3 4 6 b. Find the constants a, b, and c such that overline v =(3x+ay+z) overline i +(2x-y+bz) overline j + (x + cy + z) * kis irrotational? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A.Find the directional derivative of f(x,y,z) = xy² + 4xyz + z² at (1,2,3) in the direction of 3i + 4j -5 overline k 12 3 4 6 b. Find the constants a, b, and c such that overline v =(3x+ay+z) overline i +(2x-y+bz) overline j + (x + cy + z) * kis irrotational?.
Solutions for A.Find the directional derivative of f(x,y,z) = xy² + 4xyz + z² at (1,2,3) in the direction of 3i + 4j -5 overline k 12 3 4 6 b. Find the constants a, b, and c such that overline v =(3x+ay+z) overline i +(2x-y+bz) overline j + (x + cy + z) * kis irrotational? in English & in Hindi are available as part of our courses for UPSC.
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