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Let i and j be the unit vectors in the x and y directions, respectively. For the function F(x, y) = x3 + y3 the gradient of the function i.e.. ∇F is given by
- a)3x2 i −2yj
- b)6x2 y
- c)3x2 i+2yj
- d)2yi-3x2j
Correct answer is option 'C'. Can you explain this answer?
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Explanation:
The gradient of a function is a vector that points in the direction of the steepest increase of the function. It is defined as the vector of partial derivatives of the function with respect to each of its variables. Let's find the gradient of the given function F(x, y) = x3 y3.
Step 1: Find the partial derivative of F(x, y) with respect to x. To do that, we differentiate F(x, y) with respect to x while treating y as a constant. The result is:
F(x, y) = x3 y3
∂F/∂x = 3x2 y3
Step 2: Find the partial derivative of F(x, y) with respect to y. To do that, we differentiate F(x, y) with respect to y while treating x as a constant. The result is:
F(x, y) = x3 y3
∂F/∂y = 3x3 y2
Step 3: Combine the partial derivatives to form the gradient vector. The gradient of F(x, y) is given by:
∇F(x, y) = ∂F/∂x i + ∂F/∂y j
= 3x2 y3 i + 3x3 y2 j
= 3x2 y2 (yi + 2xj)
Therefore, the correct answer is option 'C': 3x2 i + 2yj.
The gradient of a function is a vector that points in the direction of the steepest increase of the function. It is defined as the vector of partial derivatives of the function with respect to each of its variables. Let's find the gradient of the given function F(x, y) = x3 y3.
Step 1: Find the partial derivative of F(x, y) with respect to x. To do that, we differentiate F(x, y) with respect to x while treating y as a constant. The result is:
F(x, y) = x3 y3
∂F/∂x = 3x2 y3
Step 2: Find the partial derivative of F(x, y) with respect to y. To do that, we differentiate F(x, y) with respect to y while treating x as a constant. The result is:
F(x, y) = x3 y3
∂F/∂y = 3x3 y2
Step 3: Combine the partial derivatives to form the gradient vector. The gradient of F(x, y) is given by:
∇F(x, y) = ∂F/∂x i + ∂F/∂y j
= 3x2 y3 i + 3x3 y2 j
= 3x2 y2 (yi + 2xj)
Therefore, the correct answer is option 'C': 3x2 i + 2yj.
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Let i and j be the unit vectors in the x and y directions, respectively. For the function F(x, y) = x3+ y3the gradient of the function i.e.. ∇F is given bya)3x2 i −2yjb)6x2 yc)3x2 i+2yjd)2yi-3x2jCorrect answer is option 'C'. Can you explain this answer?
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