To find the gradient of the function u = φ(r), where r = √(x² + y² + z²), we need to use the chain rule for differentiation.

Step 1: Differentiate φ(r) with respect to r to get φ′(r).

Step 2: Differentiate r with respect to x, y, and z to get the vector (x/r, y/r, z/r). This is the unit vector in the direction of r, denoted as 𝒓𝑟.

Step 3: Multiply φ′(r) by 𝒓𝑟 to get the gradient of u, which is 𝛁𝑢 = φ′(r) 𝒓𝑟.

So, the gradient of the function u = φ(r), where r = √(x² + y² + z²), is 𝛁𝑢 = φ′(r) 𝒓𝑟.

Question

To find the gradient of the function u = φ(r), where r = √(x² + y² + z²), we need to use the chain rule for differentiation.

Step 1: Differentiate φ(r) with respect to r to get φ′(r).

Step 2: Differentiate r with respect to x, y, and z to get the vector (x/r, y/r, z/r). This is the unit vector in the direction of r, denoted as 𝒓𝑟.

Step 3: Multiply φ′(r) by 𝒓𝑟 to get the gradient of u, which is 𝛁𝑢 = φ′(r) 𝒓𝑟.

So, the gradient of the function u = φ(r), where r = √(x² + y² + z²), is 𝛁𝑢 = φ′(r) 𝒓𝑟.

Knowee AI · Accepted Answer

To find the gradient of the function u = φ(r), where r = √(x² + y² + z²), we need to use the chain rule for differentiation.

Step 1: Differentiate φ(r) with respect to r to get φ′(r).

Step 2: Differentiate r with respect to x, y, and z to get the vector (x/r, y/r, z/r). This is the unit vector in the direction of r, denoted as 𝒓𝑟.

Step 3: Multiply φ′(r) by 𝒓𝑟 to get the gradient of u, which is 𝛁𝑢 = φ′(r) 𝒓𝑟.

So, the gradient of the function u = φ(r), where r = √(x² + y² + z²), is 𝛁𝑢 = φ′(r) 𝒓𝑟.

7. Find the gradient for the function 𝑢 = 𝜑(𝑟), 𝑟 = √𝑥2 + 𝑦2 + 𝑧2.Answer: 𝛁𝑢 = 𝜑′(𝑟) 𝒓𝑟 .