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Given the vector A = (cos x)(cos y)âx + (cos x)(cos y) ây, âx & ây denote unit vectors along x,y directions respectively. The curl of A is ________.
- a)(cosx cosy - sinx siny) âx
- b)(- sinx cosy + cosx siny) âz
- c)(sinx cosy - cosx siny) âz
- d)(- sinx cosy + cosx siny) ây
Correct answer is option 'B'. Can you explain this answer?
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Given the vector A = (cos x)(cos y)ax + (cos x)(cos y) ay, ax & ay...
Calculation of Curl of Vector A:
1. Given Vector A:
A = (cos x)(cos y)ax + (cos x)(cos y) ay
2. Expressing A in terms of Components:
A = (cos x)(cos y)ax + (cos x)(cos y) ay
= (cos x)(cos y)i + (cos x)(cos y)j
3. Calculating Curl of A:
The curl of a vector A in 2D is given by:
curl A = (d(A2)/dx - d(A1)/dy)az
where A = A1i + A2j
4. Calculating Partial Derivatives:
d(A2)/dx = d/dx[(cos x)(cos y)] = -sin x cos y
d(A1)/dy = d/dy[(cos x)(cos y)] = -sin y cos x
5. Substitute Partial Derivatives in Curl Formula:
curl A = (-sin x cos y - (-sin y cos x))az
= (-sin x cos y + sin y cos x)az
6. Final Answer:
Therefore, the curl of vector A is (-sin x cos y + sin y cos x)az, which matches with option B: (-sin x cos y + cos x sin y)az.
1. Given Vector A:
A = (cos x)(cos y)ax + (cos x)(cos y) ay
2. Expressing A in terms of Components:
A = (cos x)(cos y)ax + (cos x)(cos y) ay
= (cos x)(cos y)i + (cos x)(cos y)j
3. Calculating Curl of A:
The curl of a vector A in 2D is given by:
curl A = (d(A2)/dx - d(A1)/dy)az
where A = A1i + A2j
4. Calculating Partial Derivatives:
d(A2)/dx = d/dx[(cos x)(cos y)] = -sin x cos y
d(A1)/dy = d/dy[(cos x)(cos y)] = -sin y cos x
5. Substitute Partial Derivatives in Curl Formula:
curl A = (-sin x cos y - (-sin y cos x))az
= (-sin x cos y + sin y cos x)az
6. Final Answer:
Therefore, the curl of vector A is (-sin x cos y + sin y cos x)az, which matches with option B: (-sin x cos y + cos x sin y)az.
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Given the vector A = (cos x)(cos y)ax + (cos x)(cos y) ay, ax & ay...
Given
∇ × A = (- sinx cosy + cosx siny) âz
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Given the vector A = (cos x)(cos y)ax + (cos x)(cos y) ay, ax & ay denote unit vectors along x,y directions respectively. The curl of A is ________.a)(cosx cosy - sinx siny) axb)(- sinx cosy + cosx siny) azc)(sinx cosy - cosx siny) azd)(- sinx cosy + cosx siny) ayCorrect answer is option 'B'. Can you explain this answer?
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