Math Problem Statement

The problem asks to verify Gauss's law for a cube with side length 'a' centered at the origin, using four different vector fields provided in the question.

Solution

The image you uploaded is a question in Hebrew, related to vector fields. The question asks for the representation of a vector field F\mathbf{F} in different coordinate systems or forms.

Here are the main parts of the question:

  1. a. The vector field F\mathbf{F} in Cartesian coordinates.
  2. b. The vector field F\mathbf{F} in polar coordinates.
  3. c. The vector field F\mathbf{F} in cylindrical coordinates.
  4. d. The vector field F\mathbf{F} in spherical coordinates.

Additionally, the vector field is given as: F=(13x3,0,eyz)\mathbf{F} = \left( \frac{1}{3} x^3, 0, e^y z \right)

Do you need help solving this question or understanding the transformations between coordinate systems for vector fields? Would you like a detailed explanation of each part of the question?

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Math Problem Analysis

Mathematical Concepts

Vector fields
Gauss's law
Divergence theorem

Formulas

∇·F (divergence of F)
Flux through surface S = ∬_S (F · dA)
Volume integral of divergence: ∭_V (∇·F) dV

Theorems

Gauss's Divergence Theorem

Suitable Grade Level

Undergraduate Level (Physics/Engineering/Mathematics)