Question

A two-dimensional temperature profile is given as \( T = 2x^2 + 3xy + y^2 \). If \(\hat{\imath}\) and \(\hat{\jmath}\) are the unit vectors along x and y directions, respectively, which one of the following is the directional derivative of T at the location x = 2, y = 2?

• A. \( 8\hat{\imath} + 12\hat{\jmath} \)
• B. \( 14\hat{\imath} + 10\hat{\jmath} \)
• C. \( 8\hat{\imath} - 12\hat{\jmath} \)
• D. \( 14\hat{\imath} - 10\hat{\jmath} \)

Hint:

Compute partial T/partial x = 4x+3y and partial T/partial y = 3x+2y, then substitute x=2, y=2 into both.

Answer 1

The Correct Option is B

Step 1: Recall the gradient of a scalar field.

For T(x,y), the gradient vector is \( \nabla T = \frac{\partial T}{\partial x}\hat{\imath} + \frac{\partial T}{\partial y}\hat{\jmath} \).

Step 2: Differentiate.

\( \frac{\partial T}{\partial x} = 4x + 3y \), \( \frac{\partial T}{\partial y} = 3x + 2y \).

Step 3: Evaluate at (2,2).

\( 4(2)+3(2)=14 \), \( 3(2)+2(2)=10 \).

Step 4: Assemble.

\( \nabla T = 14\hat{\imath}+10\hat{\jmath} \).

\[ \boxed{\nabla T = 14\hat{\imath} + 10\hat{\jmath} \ \Rightarrow \ \text{Option (B)}} \]