[Tex]\vec{\nabla} \cdot(\nabla \times \vec{F}) [/Tex]	0
[Tex]\vec{\nabla} \times(\vec{\nabla} f) [/Tex]	0
[Tex]\vec{\nabla} \cdot(\overrightarrow{\nabla f} \times \overrightarrow{\nabla g}) [/Tex]	0
[Tex]\vec{\nabla} \cdot(f \overrightarrow{\nabla g}-g \vec{\nabla}) [/Tex]	[Tex]f \vec{\nabla}^2 g-g \vec{\nabla}^2 f [/Tex]
[Tex]\vec{\nabla} \times(\nabla \times \vec{F}) [/Tex]	[Tex]\vec{\nabla}(\vec{\nabla} \cdot \vec{F})-\vec{\nabla}^2 [/Tex]

Vector Calculus Applications

Vector Calculus or vector analysis has a number of applications in the real world:

• Navigation
• Sports
• Partial differential equation
• Three-dimensional geometry
• Used in heat transfer

Related Resources,

• Implicit Differentiation in Calculus
• How to find Derivatives?
• Vector Algebra

Solved Examples on Vector Calculus

Example 1: If F(x,y,z) = 3xy² – y²z³ then find gradF or ∇ f.

Solution:

∇ f = (∂/∂x i + ∂/∂y j + ∂/∂z k)(3xy² – y²z³)

⇒ ∇ f = ∂/∂x(3xy² – y²z³)i + ∂/∂y(3xy² – y²z³)j + ∂/∂z (3xy² – y²z³)k

⇒ ∇ f = 3y²i + (6xy – 2yz³) + (-3y²z²)k

Example 2: Find the div(F) or ∇·F, if F = xz² i – 2y²z³ j + x²yz k

Solution:

div(F) = ∇·F = (∂/∂x i + ∂/∂y j + ∂/∂z k)·(xz2 i – 2y2z3 j + x2yz k)

⇒ ∇·F = ∂/∂x(xz2) + ∂/∂y(- 2y2z3) + ∂/∂z(x2yz)

⇒ ∇·F = z² – 4yz³ + x²y

Example 3: Find curl F i.e. ∇ × f if F = xz² i – 2x³yz j + 2yz⁴ k

Solution:

∇ × f= (∂/∂x i + ∂/∂y j + ∂/∂z k) ⨯ (xz² i – 2x³yz j + 2yz⁴ k)

⇒ ∇ × f = [Tex]\begin{vmatrix} i & j & k \\ \partial /\partial x& \partial /\partial y & \partial /\partial z \\ xz^{2} & 2x^{3}yz & 2yz^{4} \\ \end{vmatrix} [/Tex]

⇒ ∇ × f = [∂/∂y(2yz⁴) – ∂/∂z(-2x³yz)]i + [∂/∂z(xz²) – ∂/∂x(2yz⁴)]j + [∂/∂x(-2x³yz) – ∂/∂y(xz²)]k

⇒ ∇ × f = [2z⁴ + 2x³y]i + [2xz – 0]j + [-6x²yz – 0]k

⇒ ∇ × f = (2z⁴ + 2x³y)i + (2xz)j -(6x²yz)k

Solved Examples

1. Calculate the gradient of f(x,y,z) = x^2y + yz^3 + xz

Solution:

∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)

∂f/∂x = 2xy + z

∂f/∂y = x^2 + z^3

∂f/∂z = 3yz^2 + x

∇f = (2xy + z, x^2 + z^3, 3yz^2 + x)

2. Find the divergence of F(x,y,z) = (x^2 + y)i + (y^2 + z)j + (z^2 + x)k

Solution:

div F = ∇ · F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z

∂F_x/∂x = 2x

∂F_y/∂y = 2y

∂F_z/∂z = 2z

div F = 2x + 2y + 2z

3. Calculate the curl of F(x,y,z) = (yz)i + (xz)j + (xy)k

Solution:

curl F = ∇ × F = (∂F_z/∂y – ∂F_y/∂z)i + (∂F_x/∂z – ∂F_z/∂x)j + (∂F_y/∂x – ∂F_x/∂y)k

∂F_z/∂y – ∂F_y/∂z = x – x = 0

∂F_x/∂z – ∂F_z/∂x = y – y = 0

∂F_y/∂x – ∂F_x/∂y = z – z = 0

curl F = 0i + 0j + 0k = 0

4. Evaluate ∫_C (x^2 + y^2) ds, where C is the circle x^2 + y^2 = 1, traversed counterclockwise.

Solution:

Parameterize the circle: x = cos(t), y = sin(t), 0 ≤ t ≤ 2π

ds = √((dx/dt)^2 + (dy/dt)^2) dt = √(sin^2(t) + cos^2(t)) dt = dt

∫_C (x^2 + y^2) ds = ∫_0^(2π) (cos^2(t) + sin^2(t)) dt = ∫_0^(2π) 1 dt = 2π

5. Compute the flux of F(x,y,z) = xi + yj + zk through the surface of the sphere x^2 + y^2 + z^2 = a^2.

Solution:

Use the divergence theorem: ∫∫_S F · dS = ∫∫∫_V div F dV

div F = 1 + 1 + 1 = 3

Volume of sphere = (4/3)πa^3

Flux = ∫∫∫_V 3 dV = 3 * (4/3)πa^3 = 4πa^3

Practice Problems on Vector Calculus in Maths

1. Calculate the gradient of f(x,y,z) = ln(x² + y² + z²)

2. Find the divergence of F(x,y,z) = (x²y)i + (y²z)j + (z²x)k

3. Compute the curl of F(x,y,z) = (y²)i + (z²)j + (x²)k

4. .Evaluate the line integral ∫_C (x + y) ds along the curve C: x = t, y = t², z = t³, 0 ≤ t ≤ 1

5. Calculate the surface integral of F(x,y,z) = xi + yj + zk over the surface z = x² + y², 0 ≤ z ≤ 1

6. Find the gradient of f(x,y) = e^(x+y) sin(xy)

7. Determine if F(x,y,z) = (yz)i + (zx)j + (xy)k is conservative

8. Compute the divergence of F(r,θ,φ) = r^ cos(φ)r + r sin(θ) sin(φ)θ + r cos(θ)φ in spherical coordinates

9. Find the potential function for F(x,y) = (2x + y)i + (x + 2y)j

10. Calculate the work done by F(x,y,z) = xi + yj + zk along the helix r(t) = (cos(t), sin(t), t), 0 ≤ t ≤ 2π

Summary

Vector calculus is a branch of mathematics that deals with vector fields in two or more dimensions. It encompasses key concepts such as the gradient, which measures the rate and direction of change in scalar fields; divergence, which quantifies the “outflow” of vector fields; and curl, which describes the rotation of vector fields. The field also includes techniques for integration over curves (line integrals) and surfaces (surface integrals). Fundamental theorems like the Divergence Theorem and Stokes’ Theorem connect these concepts, while the study of conservative fields and potential functions provides insights into path independence and field representations. These tools and concepts are essential for analyzing complex systems in physics, engineering, and other scientific disciplines, allowing for the mathematical description of phenomena involving fluid dynamics, electromagnetism, and other multidimensional processes.

FAQs on Vector Calculus

What is Vector Calculus?

Vector Calculus is branch of mathematics that deals with the differentiation and integration of Vector Function.

What is Gradient in Vector Calculus?

Gradient in Vector Calculus is rate of change of a scalar valued function in a vector space.

What is Divergence in Vector Calculus?

Divergence in Vector Calculus is the scalar of the vector operation of a function.

What is Curl in Vector Calculus?

Curl in Vector Calculus is vector of vector operation of a function.

What is Line Integral?

Line integral is the integration of the function along the lines of the curve.

What is Surface Integral?

Surface Integral of the function is the integration of the function over the whole surface.

Who invented Vector Calculus?

The concept of Vector Calculus was given by J. Willard Gibbs and Oliver Heaviside.

Is Vector Calculus same as vector analysis?

Yes, Vector Calculus is also called vector analysis.