For F = (12x², 8y², 5z²), find div F at (1,1,-3). Round the final result to 4 decimal places if needed.
Answer and explanation
10
Differentiate componentwise: div F = 24x + 16y + 10z.
Subjects
Vector calculus
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Sample exercises
For F = (11y², 7x², 0), calculate the z-component of curl F at (-6,-6,0). Round the final result to 4 decimal places if needed.
Answer and explanation
48
Compute ∂Fᵧ/∂x = 14x and ∂Fₓ/∂y = 22y, then subtract.
Let F = ∇φ with φ(x,y) = 4x² + 3y². Find the line integral of F from (0,0) to (6,4). Round the final result to 4 decimal places if needed.
Answer and explanation
192
The fundamental theorem for line integrals gives φ(6,4) − φ(0,0).
Study notes
Divergence of a polynomial vector field
Match each component to its own coordinate derivative.
UniversityDivergence of a polynomial vector field · formula reference
Match each component to its own coordinate derivative.
UniversityA component of curl
The order of subtraction sets the orientation.
UniversityA component of curl · formula reference
The order of subtraction sets the orientation.
UniversityConservative-field line integral
This applies when the vector field is the gradient of a single-valued potential on the path.
UniversityConservative-field line integral · formula reference
This applies when the vector field is the gradient of a single-valued potential on the path.
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