A scalar function V is given by V = 2xyz2. The gradient of V is given by:
\(2yz^{2}\hat{a}_x+2xz^{2}\hat{a}_y+4xyz\,\hat{a}_z\)
To determine the gradient of the scalar function \( V \), we need to compute the partial derivatives of \( V \) with respect to each of the variables \( x \), \( y \), and \( z \). The gradient of a scalar function \( V \) is given by:
\(\nabla V = \frac{\partial V}{\partial x} \hat{a}_x + \frac{\partial V}{\partial y} \hat{a}_y + \frac{\partial V}{\partial z} \hat{a}_z\)
Given: \( V = 2xyz^2 \)
- First, calculate the partial derivative with respect to \( x \): \(\frac{\partial V}{\partial x} = \frac{\partial}{\partial x} (2xyz^2) = 2yz^2\)
- Second, calculate the partial derivative with respect to \( y \): \(\frac{\partial V}{\partial y} = \frac{\partial}{\partial y} (2xyz^2) = 2xz^2\)
- Third, calculate the partial derivative with respect to \( z \): \(\frac{\partial V}{\partial z} = \frac{\partial}{\partial z} (2xyz^2) = 4xyz\)
Thus, the gradient of the function \( V \) is:
\(\nabla V = 2yz^2 \hat{a}_x + 2xz^2 \hat{a}_y + 4xyz \hat{a}_z\)
Comparing with the given options, the correct answer is:
\(2yz^{2}\hat{a}_x+2xz^{2}\hat{a}_y+4xyz\,\hat{a}_z\)
This matches with the correct option provided in the question. Hence, the correct answer is indeed option 3.
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