Gradient, Divergence, Curl. 20 mcqs on these vector with explained answers
Here are 20 MCQs on Gradient, Divergence, and Curl with explained answers based on commonly tested concepts:
1. **What is the gradient of a scalar function $$f(x, y, z)$$?**
A) Scalar
B) Vector
C) Matrix
D) Zero
**Answer:** B) Vector
**Explanation:** Gradient is a vector of partial derivatives showing direction and rate of max increase of $$f$$.
2. **The divergence of a vector field $$\vec{F}$$ is:**
A) A scalar
B) A vector
C) Zero always
D) Undefined
**Answer:** A) A scalar
**Explanation:** Divergence measures the net "outflow" at a point, producing a scalar.
3. **The curl of a vector field $$\vec{F}$$ results in:**
A) Scalar
B) Vector
C) Zero always
D) Matrix
**Answer:** B) Vector
**Explanation:** Curl measures rotational tendency, a vector orthogonal to rotation plane.
4. **The divergence of the curl of any vector field is:**
A) Always zero
B) Always one
C) Depends on the field
D) Undefined
**Answer:** A) Always zero
**Explanation:** Mathematical identity $$\nabla \cdot (\nabla \times \vec{F}) = 0$$.
5. **Curl of gradient of any scalar field is:**
A) Zero vector
B) Non-zero vector
C) Scalar
D) Undefined
**Answer:** A) Zero vector
**Explanation:** Curl of gradient is always zero: $$\nabla \times (\nabla f) = 0$$.
6. **Gradient of a function $$f(x,y,z) = x^2 + y^2 + z^2$$ at point $$(1, -1, 2)$$ is:**
A) $$(2, -2, 4)$$
B) $$(1, -1, 2)$$
C) $$(3, -3, 6)$$
D) $$(0, 0, 0)$$
**Answer:** A) $$(2, -2, 4)$$
**Explanation:** Gradient is $$\left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right) = (2x, 2y, 2z)$$.
7. **Divergence of $$\vec{F} = x\hat{i} + y\hat{j} + z\hat{k}$$ is:**
A) 1
B) 0
C) 3
D) $$x + y + z$$
**Answer:** C) 3
**Explanation:** $$\nabla \cdot \vec{F} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1 + 1 + 1 = 3$$.
8. **Curl of $$\vec{F} = y\hat{i} - x\hat{j} + 0\hat{k}$$ at $$(1, 1, 0)$$ is:**
A) $$0\hat{i} + 0\hat{j} + (-2)\hat{k}$$
B) $$0\hat{i} + 0\hat{j} + 2\hat{k}$$
C) $$2\hat{i} + 0\hat{j} + 0\hat{k}$$
D) Zero vector
**Answer:** B) $$0\hat{i} + 0\hat{j} + 2\hat{k}$$
**Explanation:** Curl components: $$\nabla \times \vec{F} = (0,0,\frac{\partial (-x)}{\partial x} - \frac{\partial y}{\partial y}) = (0,0,-1-(-1))=(0,0,2)$$.
9. **If $$f(x,y,z) = xy + yz + zx$$, $$\nabla f$$ is:**
A) $$(y+z, x+z, x+y)$$
B) $$(x+y, y+z, z+x)$$
C) $$(yz, xz, xy)$$
D) Zero vector
**Answer:** A) $$(y+z, x+z, x+y)$$
**Explanation:** Partial derivatives computed accordingly.
10. **The formula for divergence of $$\vec{F} = P\hat{i} + Q\hat{j} + R\hat{k}$$ is:**
A) $$\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$$
B) $$\nabla \times \vec{F}$$
C) $$\nabla \cdot \vec{F}$$
D) Both A and C
**Answer:** D) Both A and C
**Explanation:** Divergence is sum of partial derivatives of vector components.
11. **For $$\vec{F} = (z, x, y)$$, divergence $$\nabla \cdot \vec{F}$$ is:**
A) 3
B) 0
C) 1
D) $$x + y + z$$
**Answer:** B) 0
**Explanation:** $$\frac{\partial z}{\partial x}=0$$, $$\frac{\partial x}{\partial y}=0$$, $$\frac{\partial y}{\partial z}=0$$.
12. **Gradient of $$f = e^{2x} \cos y$$ is:**
A) $$(2e^{2x} \cos y, -e^{2x} \sin y)$$
B) $$(e^{2x} \cos y, e^{2x} \sin y)$$
C) $$(e^{x} \cos y, -e^{2x} \sin y)$$
D) Zero vector
**Answer:** A) $$(2e^{2x} \cos y, -e^{2x} \sin y)$$
**Explanation:** Partial derivative wrt $$x$$ and $$y$$.
13. **Curl of gradient of any scalar function is always:**
A) 1
B) -1
C) Zero vector
D) A non-zero vector
**Answer:** C) Zero vector
14. **Divergence can be interpreted as:**
A) Rate of flux expansion at a point
B) Rate of rotation at a point
C) Velocity vector
D) Gradient magnitude
**Answer:** A) Rate of flux expansion at a point
15. **Curl measures:**
A) Rotational tendency of a vector field
B) Rate of scalar function max increase
C) Divergence at a point
D) Total flux
**Answer:** A) Rotational tendency of a vector field
16. **If $$\vec{F} = \nabla \times \vec{A}$$, then $$\nabla \cdot \vec{F} =$$:**
A) 0
B) 1
C) $$\vec{A}$$
D) Undefined
**Answer:** A) 0
**Explanation:** Divergence of a curl is zero.
17. **For $$f(x,y,z)=3x^2 y + z^2$$, the gradient $$\nabla f$$ at $$(1,2,0)$$ is:**
A) $$(12,3,0)$$
B) $$(12,1,0)$$
C) $$(6,3,0)$$
D) $$(6,1,0)$$
**Answer:** A) $$(12,3,0)$$
**Explanation:** $$\partial f/\partial x=6xy$$, at (1,2,0) is 12, $$\partial f/\partial y=3x^2=3$$, $$\partial f/\partial z=2z=0$$.
18. **Which identity is TRUE?**
A) $$\nabla \times (\nabla f) = 0$$
B) $$\nabla \cdot (\nabla \times \vec{F}) = 0$$
C) $$\nabla \cdot (\nabla f) = \nabla^2 f$$
D) All above
**Answer:** D) All above
19. **In vector calculus, $$\nabla^2$$ is called:**
A) Gradient
B) Curl
C) Laplacian operator
D) Divergence
**Answer:** C) Laplacian operator
20. **If $$\vec{F} = (xy, yz, zx)$$, $$\nabla \cdot \vec{F}$$ is:**
A) $$x + y + z$$
B) $$y + z + x$$
C) $$y + z + x$$ (same as A)
D) $$yz + zx + xy$$
**Answer:** A) $$x + y + z$$
**Explanation:** $$\frac{\partial xy}{\partial x} = y$$, $$\frac{\partial yz}{\partial y} = z$$, $$\frac{\partial zx}{\partial z}=x$$, sum: $$x + y + z$$.
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20 more
Here are 20 more multiple-choice questions (MCQs) on Gradient, Divergence, and Curl with explained answers, gathering key concepts from vector calculus:
1. The gradient of $$f(x,y) = x^2y + y^3$$ is:
A) $$(2xy, x^2 + 3y^2)$$
B) $$(x^2, 3y^2)$$
C) $$(2x + y, 3y)$$
D) Zero vector
**Answer:** A) $$(2xy, x^2 + 3y^2)$$
**Explanation:** Partial derivatives $$\frac{\partial f}{\partial x} = 2xy$$, $$\frac{\partial f}{\partial y} = x^2 + 3y^2$$.
2. Divergence of $$\vec{F} = (x^2, y^2, z^2)$$ is:
A) $$x + y + z$$
B) $$2x + 2y + 2z$$
C) $$3$$
D) 0
**Answer:** B) $$2x + 2y + 2z$$
3. Curl of $$\vec{F} = (yz, zx, xy)$$ is:
A) $$0$$
B) $$(x-y, y-z, z-x)$$
C) $$(y-z, z-x, x-y)$$
D) $$(0,0,0)$$
**Answer:** D) $$(0,0,0)$$
**Explanation:** Curl evaluates to zero for this symmetric field.
4. The divergence of $$\nabla \times \vec{F}$$ is:
A) 0
B) 1
C) $$\vec{F}$$
D) Undefined
**Answer:** A) 0
5. For scalar field $$f = e^{xy}$$, $$\nabla f$$ is:
A) $$(ye^{xy}, xe^{xy})$$
B) $$(xe^{xy}, ye^{xy})$$
C) Zero vector
D) $$(e^{xy}, e^{xy})$$
**Answer:** A) $$(ye^{xy}, xe^{xy})$$
6. The curl of gradient of any scalar field is:
A) Zero vector
B) Identity matrix
C) A vector equal to gradient
D) None of these
**Answer:** A) Zero vector
7. What is the Laplacian $$\nabla^2 f$$ of $$f = x^2 + y^2 + z^2$$?
A) 2
B) 3
C) 6
D) 0
**Answer:** C) 6
**Explanation:** Sum of second derivatives $$2+2+2=6$$.
8. If $$\vec{F} = (xy, yz, zx)$$, what is $$\nabla \cdot \vec{F}$$ at (1,1,1)?
A) 0
B) 3
C) 2
D) 6
**Answer:** B) 3
**Explanation:** $$ \frac{\partial xy}{\partial x}= y=1 $$, $$ \frac{\partial yz}{\partial y} = z =1 $$, $$ \frac{\partial zx}{\partial z}= x=1 $$.
9. Gradient points in the direction of:
A) Maximum increase of scalar field
B) Maximum decrease
C) Minimum change
D) Random direction
**Answer:** A) Maximum increase
10. The divergence of a constant vector field is:
A) Zero
B) Constant
C) Infinity
D) Variable
**Answer:** A) Zero
11. $$\nabla \cdot ( \nabla \times \vec{F} ) =$$
A) 0
B) 1
C) $$\vec{F}$$
D) Undefined
**Answer:** A) 0
12. Curl measures the:
A) Rotational property
B) Rate of change
C) Flux density
D) Scalar field magnitude
**Answer:** A) Rotational property
13. Divergence of $$\vec{F} = (x \sin y, y \cos z, z \sin x)$$ at origin (0,0,0) is:
A) 0
B) 1
C) -1
D) Undefined
**Answer:** A) 0
14. The curl of $$\vec{F}=(z, 0, x)$$ is:
A) $$(0, 1, -1)$$
B) $$(1, -1, 0)$$
C) $$(0, -1, 1)$$
D) $$\vec{0}$$
**Answer:** A) $$(0, 1, -1)$$
15. Gradient of the scalar $$f = \ln(x^2 + y^2)$$ at $$(1,0)$$ is:
A) $$(2,0)$$
B) $$(2,2)$$
C) $$(1,1)$$
D) $$(0,0)$$
**Answer:** A) $$(2,0)$$
16. For $$f(x,y,z) = xyz$$, $$\nabla f$$ at (1,2,3) equals:
A) $$(6,3,2)$$
B) $$(2,3,1)$$
C) $$(1,2,3)$$
D) $$(3,6,2)$$
**Answer:** A) $$(6,3,2)$$
17. The curl of $$ \vec{F} = (y, -x, 0) $$ is:
A) $$ (0, 0, -2) $$
B) $$ (0, 0, 2) $$
C) $$ (0, 0, 0) $$
D) $$ (2, 2, 0) $$
**Answer:** B) $$ (0, 0, 2) $$
18. The Laplacian operator is the divergence of the:
A) Gradient
B) Curl
C) Vector field itself
D) Scalar field
**Answer:** A) Gradient
19. The vector field $$\vec{F}$$ is irrotational if:
A) $$\nabla \times \vec{F} = \vec{0}$$
B) $$\nabla \cdot \vec{F} = 0$$
C) $$\nabla f = \vec{0}$$
D) $$\vec{F} = \vec{0}$$
**Answer:** A) $$\nabla \times \vec{F} = \vec{0}$$
20. Apply divergence to $$\vec{F} = \nabla \phi$$ where $$\phi$$ is scalar. The result is:
A) Laplacian of $$\phi$$
B) Zero
C) Gradient of $$\phi$$
D) Curl of $$\phi$$
**Answer:** A) Laplacian of $$\phi$$
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20 more
Here are 20 more multiple-choice questions (MCQs) on Gradient, Divergence, and Curl with explained answers, gathering key concepts from vector calculus:
1. The gradient of $$f(x,y) = x^2y + y^3$$ is:
A) $$(2xy, x^2 + 3y^2)$$
B) $$(x^2, 3y^2)$$
C) $$(2x + y, 3y)$$
D) Zero vector
**Answer:** A) $$(2xy, x^2 + 3y^2)$$
**Explanation:** Partial derivatives $$\frac{\partial f}{\partial x} = 2xy$$, $$\frac{\partial f}{\partial y} = x^2 + 3y^2$$.
2. Divergence of $$\vec{F} = (x^2, y^2, z^2)$$ is:
A) $$x + y + z$$
B) $$2x + 2y + 2z$$
C) $$3$$
D) 0
**Answer:** B) $$2x + 2y + 2z$$
3. Curl of $$\vec{F} = (yz, zx, xy)$$ is:
A) $$0$$
B) $$(x-y, y-z, z-x)$$
C) $$(y-z, z-x, x-y)$$
D) $$(0,0,0)$$
**Answer:** D) $$(0,0,0)$$
**Explanation:** Curl evaluates to zero for this symmetric field.
4. The divergence of $$\nabla \times \vec{F}$$ is:
A) 0
B) 1
C) $$\vec{F}$$
D) Undefined
**Answer:** A) 0
5. For scalar field $$f = e^{xy}$$, $$\nabla f$$ is:
A) $$(ye^{xy}, xe^{xy})$$
B) $$(xe^{xy}, ye^{xy})$$
C) Zero vector
D) $$(e^{xy}, e^{xy})$$
**Answer:** A) $$(ye^{xy}, xe^{xy})$$
6. The curl of gradient of any scalar field is:
A) Zero vector
B) Identity matrix
C) A vector equal to gradient
D) None of these
**Answer:** A) Zero vector
7. What is the Laplacian $$\nabla^2 f$$ of $$f = x^2 + y^2 + z^2$$?
A) 2
B) 3
C) 6
D) 0
**Answer:** C) 6
**Explanation:** Sum of second derivatives $$2+2+2=6$$.
8. If $$\vec{F} = (xy, yz, zx)$$, what is $$\nabla \cdot \vec{F}$$ at (1,1,1)?
A) 0
B) 3
C) 2
D) 6
**Answer:** B) 3
**Explanation:** $$ \frac{\partial xy}{\partial x}= y=1 $$, $$ \frac{\partial yz}{\partial y} = z =1 $$, $$ \frac{\partial zx}{\partial z}= x=1 $$.
9. Gradient points in the direction of:
A) Maximum increase of scalar field
B) Maximum decrease
C) Minimum change
D) Random direction
**Answer:** A) Maximum increase
10. The divergence of a constant vector field is:
A) Zero
B) Constant
C) Infinity
D) Variable
**Answer:** A) Zero
11. $$\nabla \cdot ( \nabla \times \vec{F} ) =$$
A) 0
B) 1
C) $$\vec{F}$$
D) Undefined
**Answer:** A) 0
12. Curl measures the:
A) Rotational property
B) Rate of change
C) Flux density
D) Scalar field magnitude
**Answer:** A) Rotational property
13. Divergence of $$\vec{F} = (x \sin y, y \cos z, z \sin x)$$ at origin (0,0,0) is:
A) 0
B) 1
C) -1
D) Undefined
**Answer:** A) 0
14. The curl of $$\vec{F}=(z, 0, x)$$ is:
A) $$(0, 1, -1)$$
B) $$(1, -1, 0)$$
C) $$(0, -1, 1)$$
D) $$\vec{0}$$
**Answer:** A) $$(0, 1, -1)$$
15. Gradient of the scalar $$f = \ln(x^2 + y^2)$$ at $$(1,0)$$ is:
A) $$(2,0)$$
B) $$(2,2)$$
C) $$(1,1)$$
D) $$(0,0)$$
**Answer:** A) $$(2,0)$$
16. For $$f(x,y,z) = xyz$$, $$\nabla f$$ at (1,2,3) equals:
A) $$(6,3,2)$$
B) $$(2,3,1)$$
C) $$(1,2,3)$$
D) $$(3,6,2)$$
**Answer:** A) $$(6,3,2)$$
17. The curl of $$ \vec{F} = (y, -x, 0) $$ is:
A) $$ (0, 0, -2) $$
B) $$ (0, 0, 2) $$
C) $$ (0, 0, 0) $$
D) $$ (2, 2, 0) $$
**Answer:** B) $$ (0, 0, 2) $$
18. The Laplacian operator is the divergence of the:
A) Gradient
B) Curl
C) Vector field itself
D) Scalar field
**Answer:** A) Gradient
19. The vector field $$\vec{F}$$ is irrotational if:
A) $$\nabla \times \vec{F} = \vec{0}$$
B) $$\nabla \cdot \vec{F} = 0$$
C) $$\nabla f = \vec{0}$$
D) $$\vec{F} = \vec{0}$$
**Answer:** A) $$\nabla \times \vec{F} = \vec{0}$$
20. Apply divergence to $$\vec{F} = \nabla \phi$$ where $$\phi$$ is scalar. The result is:
A) Laplacian of $$\phi$$
B) Zero
C) Gradient of $$\phi$$
D) Curl of $$\phi$$
**Answer:** A) Laplacian of $$\phi$$
***
