Question

check the fundamental theorem for gradients using = ଶ 4 2ଷ, the points a =(0,0,0), b = (1,1,1), and the following three paths:

Answer 1

To check the fundamental theorem for gradients, we need to use the formula ∇f = ∮ grad(f) · ds. Given the vector field F = (4,2) and the points a = (0,0) and b = (1,1), we can calculate the line integral along three different paths using different parameterizations and compare with the gradients of scalar functions.

Step-by-step explanation:

To check the fundamental theorem for gradients, we need to use the formula:

∊f = ∫ grad(f) ⋅ ds

Given the vector field F = (4,2) and the points a = (0,0) and b = (1,1), we can calculate the line integral along the three paths using different parametrizations:

• Path 1: Straight line from a to b: Parameterization - r(t) = (t,t) for 0 ≤ t ≤ 1
• Path 2: Quarter-circle from a to b: Parameterization - r(t) = (cos(t),sin(t)) for 0 ≤ t ≤ π/2
• Path 3: Sine wave from a to b: Parameterization - r(t) = (t,sin(2πt)) for 0 ≤ t ≤ 1

By calculating the line integrals using these parameterizations, we can then calculate the gradients of the scalar functions and compare with the vector field F to check if the fundamental theorem holds.