Question

Let F(x, y, z) = (8x₇ ln(3y²+6) + 10z⁸)i + (6yx⁸ / (3y²+6) + 7z)j + (80x z⁷ + 7y - 6π sin(πz))k be a vector field.

a) Find ∇ · F.
b) Compute ∇ × F.
c) Determine if F is conservative. If it is, find a potential function f such that F = ∇f.

Answer 1

The student's question involves high school-level physics, specifically the application of vector operations like dot and cross product for understanding properties of forces, work, and vector fields.

Step-by-step explanation:

The student is asking about various physics concepts involving vectors, such as calculating angles between vectors and the axes, demonstrating orthogonality between force vectors, vector products, work done by forces, and electric fields. Part of these questions involves using the vector operations dot product (for orthogonality checks and work calculations) and cross product (for finding vector products).

For example, to show two force vectors are orthogonal, one would take the dot product: Ď · Ğ. If the result is zero, then the vectors are orthogonal. To calculate work done by a force when a particle moves along a path, the dot product F · d is used, where F is the force and d is the displacement vector.

For the vector field problems, one should use the divergence (∇ · F) and curl (∇ × F) operators to find the respective vector field properties. Conservative fields are identified if the curl of the vector field is zero, and in that case, a potential function that satisfies F = ∇f should be found.