vector calculus solved problems

Collapse menu

Introduction, 1 analytic geometry.

• 2. Distance Between Two Points; Circles
• 3. Functions
• 4. Shifts and Dilations

2 Instantaneous Rate of Change: The Derivative

• 1. The slope of a function
• 2. An example
• 4. The Derivative Function
• 5. Properties of Functions

3 Rules for Finding Derivatives

• 1. The Power Rule
• 2. Linearity of the Derivative
• 3. The Product Rule
• 4. The Quotient Rule
• 5. The Chain Rule

4 Transcendental Functions

• 1. Trigonometric Functions
• 2. The Derivative of $\sin x$
• 3. A hard limit
• 4. The Derivative of $\sin x$, continued
• 5. Derivatives of the Trigonometric Functions
• 6. Exponential and Logarithmic functions
• 7. Derivatives of the exponential and logarithmic functions
• 8. Implicit Differentiation
• 9. Inverse Trigonometric Functions
• 10. Limits revisited
• 11. Hyperbolic Functions

5 Curve Sketching

• 1. Maxima and Minima
• 2. The first derivative test
• 3. The second derivative test
• 4. Concavity and inflection points
• 5. Asymptotes and Other Things to Look For

6 Applications of the Derivative

• 1. Optimization
• 2. Related Rates
• 3. Newton's Method
• 4. Linear Approximations
• 5. The Mean Value Theorem

7 Integration

• 1. Two examples
• 2. The Fundamental Theorem of Calculus
• 3. Some Properties of Integrals

8 Techniques of Integration

• 1. Substitution
• 2. Powers of sine and cosine
• 3. Trigonometric Substitutions
• 4. Integration by Parts
• 5. Rational Functions
• 6. Numerical Integration
• 7. Additional exercises

9 Applications of Integration

• 1. Area between curves
• 2. Distance, Velocity, Acceleration
• 4. Average value of a function
• 6. Center of Mass
• 7. Kinetic energy; improper integrals
• 8. Probability
• 9. Arc Length
• 10. Surface Area

10 Polar Coordinates, Parametric Equations

• 1. Polar Coordinates
• 2. Slopes in polar coordinates
• 3. Areas in polar coordinates
• 4. Parametric Equations
• 5. Calculus with Parametric Equations

11 Sequences and Series

• 1. Sequences
• 3. The Integral Test
• 4. Alternating Series
• 5. Comparison Tests
• 6. Absolute Convergence
• 7. The Ratio and Root Tests
• 8. Power Series
• 9. Calculus with Power Series
• 10. Taylor Series
• 11. Taylor's Theorem
• 12. Additional exercises

12 Three Dimensions

• 1. The Coordinate System
• 3. The Dot Product
• 4. The Cross Product
• 5. Lines and Planes
• 6. Other Coordinate Systems

13 Vector Functions

• 1. Space Curves
• 2. Calculus with vector functions
• 3. Arc length and curvature
• 4. Motion along a curve

14 Partial Differentiation

• 1. Functions of Several Variables
• 2. Limits and Continuity
• 3. Partial Differentiation
• 4. The Chain Rule
• 5. Directional Derivatives
• 6. Higher order derivatives
• 7. Maxima and minima
• 8. Lagrange Multipliers

15 Multiple Integration

• 1. Volume and Average Height
• 2. Double Integrals in Cylindrical Coordinates
• 3. Moment and Center of Mass
• 4. Surface Area
• 5. Triple Integrals
• 6. Cylindrical and Spherical Coordinates
• 7. Change of Variables

16 Vector Calculus

1. vector fields, 2. line integrals, 3. the fundamental theorem of line integrals, 4. green's theorem, 5. divergence and curl, 6. vector functions for surfaces, 7. surface integrals, 8. stokes's theorem, 9. the divergence theorem, 17 differential equations.

• 1. First Order Differential Equations
• 2. First Order Homogeneous Linear Equations
• 3. First Order Linear Equations
• 4. Approximation
• 5. Second Order Homogeneous Equations
• 6. Second Order Linear Equations
• 7. Second Order Linear Equations, take two

18 Useful formulas

19 introduction to sage.

• 2. Differentiation
• 3. Integration

5.1 Vector Calculus

Extend multivariable calculus to vector fields, then apply your new skills by exploring Maxwell's equations.

Vector Calculus in a Nutshell

Explore the possibilities that come from combining calculus and vectors.

End of Unit 1

Complete all lessons above to reach this milestone.

0 of 1 lessons complete

Calculus of Motion

Look at the world in motion through the lens of vector calculus.

• Space Curves

Fly along curves through three dimensions.

Integrals and Arc Length

Exactly how long is a space curve?

Frenet Formulae

Measure the shape of space curves with vector calculus.

• Parametric Surfaces

Expand your gallery of shapes to include a few exotic cases.

• Vector Fields

Combine vectors and functions into a powerful tool for applications.

Jack and the Beanstalk

Learn about Newton's root-finding method and help defeat a vile giant.

Electrostatic Bootcamp

Take a lightning tour of the physics that made vector calculus famous.

End of Unit 2

0 of 8 lessons complete

Introducing Surface Integrals

Mix vectors with integrals to uncover an essential tool for applications.

Flux (Part I)

Experiment with charges moving in electric field and discover the concept of flux.

Flux (Part II)

Use flux to uncover surface integrals and see how they're used to solve important problems.

• Surface Integrals

Master integrals of functions on parametrized surfaces.

Divergence (Part I)

Construct a crucial vector derivative through a fundamental law of nature.

Divergence (Part II)

What is divergence?

The Divergence Theorem

Prove a beautiful integral identity that's essential for real-world applications.

The Divergence Theorem and Fluids

Another view of one of the most important theorems in vector calculus.

Flows & Divergence

Explore more about the divergence through visuals and geometry.

End of Unit 3

0 of 9 lessons complete

Work (Part I)

Explore an important physics application of vector calculus.

Work (Part II)

Unveil a new kind of integral by delving into a familiar physics concept.

• Line Integrals

Learn how to integrate along space curves and why it's so useful.

Path Independence

Journey to where calculus and topology meet to discover a crucial property of vector fields.

Construct a derivative that measures the swirl of a vector field.

• Stokes' Theorem

Uncover the deep connection between curl and line integrals.

Swirls and Curls

Dive deeper into curl with visuals and geometry.

Differential Forms (Optional)

Unify vector calculus into a single master formula.

End of Unit 4

The Laplacian

Apply a new derivative to problems in electrostatics and fluid dynamics.

Gaussian Integrals (Part I)

Detour into the world of multivariable calculus to compute an integral crucial for applications.

Gaussian Integrals (Part II)

Use linear algebra and a change of variables to find the most general Gaussian integral.

• Fourier Transform

Apply Gaussian integrals to understand the Fourier transform, a powerful way to solve pdes.

The Diffusion Equation

Use the Fourier transform to solve a pde and calculate the speed of scent.

The Wave Equation

Make waves with Fourier series.

Maxwell's Equations (Part I)

Combine divergence, curl, line and surface integrals to uncover Maxwell's electromagnetic equations.

Maxwell's Equations (Part II)

Use vector calculus to solve electromagnetic problems and unify Maxwell's equations with forms.

More Electrostatics!

Squeeze new pde methods from old vector calculus theorems and solve hard electrostatics problems.

End of Unit 5

Course description.

Change is deeply rooted in the natural world. Fluids, electromagnetic fields, the orbits of planets, the motion of molecules; all are described by vectors and all have characteristics depending on where we look and when. In this course, you'll learn how to quantify such change with calculus on vector fields. Go beyond the math to explore the underlying ideas scientists and engineers use every day.

Topics covered

• Differential Forms
• Divergence Theorem
• Maxwell's Equations

Prerequisites and next steps

You’ll need a solid understanding of algebra, as well as the basics of functions, limits, derivatives, and integrals.

Prerequisites

• Multivariable Calculus

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons

Margin Size

• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

4.5: Path Independence, Conservative Fields, and Potential Functions

• Last updated
• Save as PDF
• Page ID 568

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

For certain vector fields, the amount of work required to move a particle from one point to another is dependent only on its initial and final positions, not on the path it takes. Gravitational and electric fields are examples of such vector fields. This section will discuss the properties of these vector fields.

Definition: Path Independent and Conservative

Let \(\mathbf{F}\) be a vector field defined on an open region D in space, and suppose that for any two points A and B in D the line integral

\[\int_{C}^{ }\mathbf{F}\cdot \mathit{d}\mathbf{r}\]

along a path C from A to B in D is the same over all paths from A to B . Then the integral \[\int_{C}^{ }\mathbf{F}\cdot \mathit{d}\mathbf{r}\] is path independent in D and the field F is conservative on D .

Potential Function

Definition: If F is a vector field defined on D and \[\mathbf{F}=\triangledown f\] for some scalar function f on D , then f is called a potential function for F . You can calculate all the line integrals in the domain F over any path between A and B after finding the potential function f

\[\int_{A}^{B}\mathbf{F}\cdot \mathit{d}\mathbf{r}=\int_{A}^{B}\triangledown f\mathit{d}\mathbf{r}=\mathit{f(B)}-\mathit{f(A)}\]

This can be related back to the Fundamental Theorem of Calculus, since the gradient can be thought of as similar to the derivative. Another important property of conservative vector fields is that the integral of F around any closed path D is always 0.

Assumptions on Curves, Vector Fields, and Domains

For computational sake, we have to assume the following properties regarding the curves, surfaces, domains, and vector fields:

• The curves we consider are piecewise smooth , meaning they are composed of many infinitesimally small, smooth pieces connected end to end.
• We assume that the domain D is a simply connected open region , meaning that any two points in D can be joined by a smooth curve within the region and that every loop in D can be contracted to a point in D without ever leaving D .

Theorem 1: Fundamental Theorem of Line Integrals

Let C be a smooth curve joining the point A to point B in the plane ore in space and parametrized by \(\mathbf{r}(t)\). Let f be a differentiable function with a continuous gradient vector \(\mathbf{F}=\bigtriangledown{f}\) on a domain D containing C . Then \(\int_{C}\mathbf{F}\cdot d\mathbf{r}=f(B)-f(A)\).

Suppose that A and B are two points in region D and that the curve C is given by \[\mathbf{r}(t)=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\] is a smooth curve in D that joins points A and B . Along C , f is a differentiable function of t and

\[\begin{align*} \dfrac{\partial f }{\partial t}&=\dfrac{\partial f }{\partial x}\dfrac{\partial x }{\partial t}+\dfrac{\partial f }{\partial y}\dfrac{\partial y }{\partial t}+\dfrac{\partial f }{\partial z}\dfrac{\partial z }{\partial t} \\ &=\bigtriangledown f\cdot \left ( \dfrac{\mathrm{d} x}{\mathrm{d} t}\mathbf{i}+\dfrac{\mathrm{d} y}{\mathrm{d} t}\mathbf{j}\dfrac{\mathrm{d} z}{\mathrm{d} t}\mathbf{k} \right ) \\ &=\bigtriangledown f\cdot \dfrac{\mathrm{d} \mathbf{r}}{\mathrm{d} t} \\ &=\mathbf{F}\cdot \dfrac{\mathrm{d} \mathbf{r}}{\mathrm{d} t} \end{align*}\]

\[\int_{C}\mathbf{F}\cdot d\mathbf{r}=\int_{t=a}^{t=b}\mathbf{F}\cdot \dfrac{\mathrm{d} \mathbf{r}}{\mathrm{d} t}dt=\int_a^b\dfrac{\mathrm{d} f}{\mathrm{d} t}dt \nonumber\]

\[\mathbf{r}(a)=A, \; \mathbf{r}(b)=B \nonumber\]

Which means:

\[\left. f(g(t),h(t),k(t))\right|_a^b=f(B)-f(A) \nonumber\]

Thus proving Theorem 1. This shows us that the integral of a gradient field is easy to compute, provided we know the function \(f\).

\(\square\)

As mentioned earlier, this is very similar to the Fundamental Theorem of Calculus both in theory and importance. Like the FTC, it provides us with a way to evaluate line integrals without limits of Riemann sums.

Theorem 2: Conservative Fields are Gradient Fields

Let \(\mathbf{F}=M\hat{\mathbf{i}}+N\hat{\mathbf{j}}+P\hat{\mathbf{k}}\) be a vector field whose components are continuous throughout an open connected region D in space. Then F is conservative if and only it F is a gradient field \(\bigtriangledown f\) for a differentiable function f.

If F is a gradient field, then \(\mathbf{F}=\bigtriangledown f\) for a differentiable function f. By Theorem 1, we know that \[\int_C\mathbf{F}\cdot d\mathbf{r}=f(B)-f(A)\] and that the value of the line integral depends only on the two endpoints, not on the path. The line integral is said to be independent and F is a conservative field.

However, suppose F is a conservative vector field and we want to find some function f on D such that \(\bigtriangledown f=\mathbf{F}\). First, we must pick a point A in the domain D such that \(f(A)=0\). For any other point B , we must define \(f(B)\) as equal to \[\int_C\mathbf{F}\cdot d\mathbf{r},\] where the curve C is any smooth path in D from A to B . Because F is conservative, we know that \(f(B)\) is not dependant on C and vice versa. In order to show that \(\bigtriangledown f=\mathbf{F},\) we need to show that

\[\dfrac{\partial f}{\partial x}=M, \dfrac{\partial f}{\partial y}=N, \dfrac{\partial f}{\partial z}=P.\nonumber\]

Suppose B has coordinates \((x,y,z)\) and a nearby point \(B_0=(x_0,y,z).\) By definition, then, the value of function f at the nearby point is \[\int_{C_0}\mathbf{F}\cdot d\mathbf{r},\] where \(C_0\) is any path from A to \(B_0.\) We can take path C to be the union between path \(C_0\) and line segment L from B to \(B_0\). Therefore,

\[f(x,y,z)=\int_{C_0}\mathbf{F}\cdot d\mathbf{r}+\int_L\mathbf{F}\cdot d\mathbf{r}\nonumber\]

We can differentiate this integral, arriving at:

\[\dfrac{\partial }{\partial x}f(x,y,z)=\dfrac{\partial}{\partial x}\left ( \int_{C_0}\mathbf{F}\cdot d\mathbf{r}+\int_L\mathbf{F}\cdot d\mathbf{r} \right ) \nonumber\]

Only the last term of the above equation is dependent on x, so

\[\dfrac{\partial }{\partial x}f(x,y,z)=\dfrac{\partial }{\partial x}\int_L\mathbf{F}\cdot d\mathbf{r} \nonumber\]

Now, if we parametrize \(L\) such that

\[\mathbf{r}(t)=t\mathbf{i}+y\mathbf{j}+z\mathbf{k} \nonumber\]

where \(x_0\leq t \leq x\) Then,

\[\dfrac{\mathrm{d} r}{\mathrm{d} t}=\mathbf{i}\] \[\mathbf{F}\cdot \dfrac{\mathrm{d} r}{\mathrm{d} t}=M \nonumber\]

\[\int_L\mathbf{f}\cdot d\mathbf{r}=\int_{x_0}^xM(t,y,z)dt] \nonumber\]

Substitution gives us

\[\dfrac{\partial }{\partial x}f(x,y,z)=\dfrac{\partial }{\partial x}\int_{x_0}^xM(t,y,z)dt=M(x,y,z) \nonumber\]

by the FTC. The partial derivatives

\[\dfrac{\partial f}{\partial y}=N \nonumber\]

\[\dfrac{\partial f}{\partial z}=P \nonumber\]

follow similarly, showing that

\[\mathbf{F}=\bigtriangledown f \nonumber\]

In other words, \(\mathbf{F}=\bigtriangledown f\) is only true when, for any two point A and B in the region D , \(\int_C\mathbf{F}\cdot d\mathbf{r}\) is independent of the path C that joins the two points in D .

Theorem 3: Looper Property of Conservative Fields

The following statements are equivalent:

• \(\oint_{C}\mathbf{F}\cdot d\mathbf{r}=0\) around every loop (closed curve C ) in D.
• The field F is conservative on D .

We want to show that for any two points A and B in D, the ingtegral of

\[\mathbf{F}\cdot d\mathbf{r}\nonumber\]

has the same value over any two paths \(C_1\) & \(C_2\) from A to B .

We reverse the direction of \(C_2\) to make the path \(-C_2\) from B to A .

Together, the two curves \(C_1\) & \(-C_2\) make a closed loop, which we will call C .

If you recall from earlier in this section, the integral over a closed loop for a conservative field is always 0:

\[\begin{align*} \int_{C_1}\mathbf{F}\cdot d\mathbf{r}-\int_{C_2}\mathbf{F}\cdot d\mathbf{r}&=\int_{C_1}\mathbf{F}\cdot d\mathbf{r}+\int_{-C_2}\mathbf{F}\cdot d\mathbf{r} \\ &=\int_C\mathbf{F}\cdot d\mathbf{r} \\ &=0 \end{align*}\]

Therefore, the integrals over \(C_1\) & \(C_2\) must be equal.

We want to show that the integral over \(\mathbf{F}\cdot d\mathbf{r}\) is zero for any closed loop C . We pick two points A & B on C and use them to break C into 2 pieces: \(C_1\) from A to B and \(C_2\) from B back to A.

\[\begin{align*} \oint _C\mathbf{F}\cdot d\mathbf{r}&=\int_{C_1}\mathbf{F}\cdot d\mathbf{r}+\int_{C_2}\mathbf{F}\cdot d\mathbf{r} \\ &=\int_A^B\mathbf{F}\cdot d\mathbf{r}-\int_A^B\mathbf{F}\cdot d\mathbf{r} \\ &=0 \end{align*}\]

Finding Potentials for Conservative Fields

Component Test for Conservative Fields : Let \(\mathbf{F}=M(x,y,z)\hat{\textbf{i}} + N(x,y,z) \hat{\textbf{j}}+ P(x,y,z) \hat{\textbf{k}} \) be a field on a connected and simply connected domain whose component functions have continuous first partial derivatives. Then, F is conservative if and only if

\[\dfrac{\partial P }{\partial x}=\dfrac{\partial M}{\partial z}\]

\[\dfrac{\partial P }{\partial y}=\dfrac{\partial N}{\partial z}\]

\[\dfrac{\partial N }{\partial x}=\dfrac{\partial M}{\partial y}.\]

*Note: See Example 2

Definition: Exact Differential Forms

Any expression

\[M(x,y,z)dx+N(x,y,z)dy+P(x,y,z)dz\]

is a differential form . A differential form is exact on a domain D in space if

\[M\,dx+N\,dy+P\,dz=\dfrac{\partial f}{\partial x}dx+\dfrac{\partial f}{\partial y}dy+\dfrac{\partial f}{\partial z}dz=df\]

for some scalar function f throughout D .

Component Test for Exactness of \(Mdx+Ndy+Pdz\): The differential form \(Mdx+Ndy+Pdz\) is exact on a connected and simply connected domain if and only if

\[\dfrac{\partial N }{\partial x}=\dfrac{\partial M}{\partial y}\]

Notice, this is the same as saying the field \(\mathbf{F}=M\hat{\mathbf{i}}+N\hat{\mathbf{j}}+P\hat{\mathbf{k}}\) is conservative.

Example \(\PageIndex{1}\)

Suppose the force field \(\mathbf{F}=\bigtriangledown f\) is the gradient of the function \(f(x,y,z)=-\dfrac{1}{x^2+y^2+z^2}\). Find the work done by F in moving an object along a smooth curve C joining \((1,0,0)\) to \((0,0,2)\) that does not pass through the origin.

Since we know that this is a conservative field, we can apply Theorem 1, which shows that regardless of the curve C , the work done by F will be as follows:

\[\begin{align*} \int_{C}\mathbf{F}\cdot d\mathbf{r}&=f(0,0,2)-f(1,0,0) \\ &=-\dfrac{1}{4}-(-1) \\ &=\dfrac{3}{4} \end{align*}\]

Example \(\PageIndex{2}\)

\[\mathbf{F}=(e^x \cos y+yz)\hat{\mathbf{i}}+(xz-e^x\sin y)\hat{\mathbf{j}}+(xy+z)\hat{\mathbf{k}} \nonumber\]

is conservative over its natural domain and find a potential function for it.

The natural domain of F is all of space, which is connected and simply connected. Let's define the following:

\[M=e^x\cos y+yz \nonumber\]

\[N=xz-e^x\sin y \nonumber\]

\[P=xy+z \nonumber\]

and calculate

\[\dfrac{\partial P }{\partial x}=y=\dfrac{\partial M}{\partial z} \nonumber\]

\[\dfrac{\partial P }{\partial y}=x=\dfrac{\partial N}{\partial z} \nonumber\]

\[\dfrac{\partial N }{\partial x}=-e^x\sin y=\dfrac{\partial M}{\partial y}. \nonumber\]

Because the partial derivatives are continuous, F is conservative. Now that we know there exists a function f where the gradient is equal to F , let's find f.

\[\dfrac{\partial f }{\partial x}=e^x\cos y+yz \nonumber\]

\[\dfrac{\partial f }{\partial y}=xz-e^x\sin y \nonumber\]

\[\dfrac{\partial f }{\partial z}=xy+z\nonumber \]

If we integrate the first of the three equations with respect to x, we find that

\[f(x,y,z)=\int(e^x/cos y+yz)dx=e^x\cos y+xyz+g(y,z)\]

where \(g(y,z)\) is a constant dependent on \(y\) and \(z\) variables. We then calculate the partial derivative with respect to \(y\) from this equation and match it with the equation of above.

\[\dfrac{\partial }{\partial y}(f(x,y,z))=-e^x/sin y+xz+\dfrac{\partial g}{\partial y}=xz-e^x\sin y \nonumber\]

This means that the partial derivative of \(g\) with respect to \(y\) is 0, thus eliminating \(y\) from \(g\) entirely and leaving it as a function of \(z\) alone.

\[f(x,y,z)=e^x\cos y+xyz+h(z) \nonumber\]

We then repeat the process with the partial derivative with respect to \(z\).

\[\dfrac{\partial }{\partial z}(f(x,y,z))=xy+\dfrac{\mathrm{d} h}{\mathrm{d} z}=xy+z \nonumber\]

which means that

\[\dfrac{\mathrm{d} h}{\mathrm{d} z}= z \nonumber\]

so we can find \(h(z)\) by integrating:

\[h(z)=\dfrac{z^2}{2}+C. \nonumber\]

\[f(x,y,z)=e^x\cos y+xyz+\dfrac{z^2}{2}+C. \nonumber\]

We still have infinitely many potential functions for F -one at each value of C.

Example \(\PageIndex{3}\)

Show that \( ydx+xdy+4dz\) is exact and evaluate the integral

\[\int_{(1,1,1)}^{(2,3,-1)}ydx+xdy+4dz \nonumber\]

over any path from \((1,1,1)\) to \((2,3,-1)\).

We let \(M=y\), \(N=x\), and \(P=4\). Apply the Test for Exactness:

\[\dfrac{\partial N}{\partial x}=1=\dfrac{\partial M}{\partial y}\nonumber\]

\[\dfrac{\partial N}{\partial z}=0=\dfrac{\partial P}{\partial y}\nonumber\]

\[\dfrac{\partial N}{\partial z}=0=\dfrac{\partial P}{\partial y}.\nonumber\]

This proves that \(ydx+xdy+4dz\) is exact, so

\[ydx+xdy+4dz=df \nonumber\]

for some function f , and the integral's value is \(f(2,3,-1)-f(1,1,1)\).

We find fo up to a constant by integrating the following equations:

\[\dfrac{\partial f}{\partial x}=y, \dfrac{\partial f}{\partial y}=x, \dfrac{\partial f}{\partial z}=4 \nonumber\]

From the first equation, we get that \(f(x,y,z)=xy+g(y,z)\)

The second equation tells us that \(\dfrac{\partial f}{\partial y}=x+\dfrac{\partial g}{\partial y}=x\)

\[\dfrac{\partial g}{\partial y}=0 \nonumber\]

\[f(x,y,z)=xy+h(z) \nonumber\]

The third equation tells us that \(\dfrac{\partial f}{\partial z}=0+\dfrac{d h}{d z}=4\) so \(h(z)=4z+C\)

\[f(x,y,z)=xy+4z+C \nonumber\]

By substitution, we find that:

\[f(2,3,-1)-f(1,1,1)=2+C-(5+C)=-3 \nonumber\]

• Weir, Maurice D., Joel Hass, and George B. Thomas. Thomas' Calculus: Early Transcendentals . Boston: Addison-Wesley, 2010. Print.

Contributors and Attributions

• Alagu Chidambaram (UCD)

Integrated by Justin Marshall.

• Science & Math
• Mathematics

Enjoy fast, free delivery, exclusive deals, and award-winning movies & TV shows with Prime Try Prime and start saving today with fast, free delivery

Amazon Prime includes:

Fast, FREE Delivery is available to Prime members. To join, select "Try Amazon Prime and start saving today with Fast, FREE Delivery" below the Add to Cart button.

• Cardmembers earn 5% Back at Amazon.com with a Prime Credit Card.
• Unlimited Free Two-Day Delivery
• Streaming of thousands of movies and TV shows with limited ads on Prime Video.
• A Kindle book to borrow for free each month - with no due dates
• Listen to over 2 million songs and hundreds of playlists
• Unlimited photo storage with anywhere access

Important:  Your credit card will NOT be charged when you start your free trial or if you cancel during the trial period. If you're happy with Amazon Prime, do nothing. At the end of the free trial, your membership will automatically upgrade to a monthly membership.

Buy new: .savingPriceOverride { color:#CC0C39!important; font-weight: 300!important; } .reinventMobileHeaderPrice { font-weight: 400; } #apex_offerDisplay_mobile_feature_div .reinventPriceSavingsPercentageMargin, #apex_offerDisplay_mobile_feature_div .reinventPricePriceToPayMargin { margin-right: 4px; } $14.88 $ 14 . 88 FREE delivery Thursday, June 13 on orders shipped by Amazon over $35 Ships from: Amazon.com Sold by: Amazon.com

Return this item for free.

Free returns are available for the shipping address you chose. You can return the item for any reason in new and unused condition: no shipping charges

• Go to your orders and start the return
• Select your preferred free shipping option
• Drop off and leave!

Save with Used - Good .savingPriceOverride { color:#CC0C39!important; font-weight: 300!important; } .reinventMobileHeaderPrice { font-weight: 400; } #apex_offerDisplay_mobile_feature_div .reinventPriceSavingsPercentageMargin, #apex_offerDisplay_mobile_feature_div .reinventPricePriceToPayMargin { margin-right: 4px; } $11.16 $ 11 . 16 FREE delivery Friday, June 14 on orders shipped by Amazon over $35 Ships from: Amazon Sold by: Jenson Books Inc

Download the free Kindle app and start reading Kindle books instantly on your smartphone, tablet, or computer - no Kindle device required .

Read instantly on your browser with Kindle for Web.

Using your mobile phone camera - scan the code below and download the Kindle app.

Image Unavailable

• To view this video download Flash Player

Follow the author

Understanding Vector Calculus: Practical Development and Solved Problems (Dover Books on Mathematics)

Purchase options and add-ons.

• ISBN-10 0486835901
• ISBN-13 978-0486835907
• Publisher Dover Publications
• Publication date February 12, 2020
• Language English
• Dimensions 6 x 0.25 x 9 inches
• Print length 112 pages
• See all details

Frequently bought together

Similar items that may ship from close to you

Editorial Reviews

About the author, product details.

• Publisher ‏ : ‎ Dover Publications (February 12, 2020)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 112 pages
• ISBN-10 ‏ : ‎ 0486835901
• ISBN-13 ‏ : ‎ 978-0486835907
• Item Weight ‏ : ‎ 6 ounces
• Dimensions ‏ : ‎ 6 x 0.25 x 9 inches
• #977 in Calculus (Books)
• #3,709 in Applied Mathematics (Books)
• #42,592 in Unknown

About the author

Jerrold franklin.

Discover more of the author’s books, see similar authors, read author blogs and more

Customer reviews

Customer Reviews, including Product Star Ratings help customers to learn more about the product and decide whether it is the right product for them.

To calculate the overall star rating and percentage breakdown by star, we don’t use a simple average. Instead, our system considers things like how recent a review is and if the reviewer bought the item on Amazon. It also analyzed reviews to verify trustworthiness.

No customer reviews

• Amazon Newsletter
• About Amazon
• Accessibility
• Sustainability
• Press Center
• Investor Relations
• Amazon Devices
• Amazon Science
• Sell on Amazon
• Sell apps on Amazon
• Supply to Amazon
• Protect & Build Your Brand
• Become an Affiliate
• Become a Delivery Driver
• Start a Package Delivery Business
• Advertise Your Products
• Self-Publish with Us
• Become an Amazon Hub Partner
• › See More Ways to Make Money
• Amazon Visa
• Amazon Store Card
• Amazon Secured Card
• Amazon Business Card
• Shop with Points
• Credit Card Marketplace
• Reload Your Balance
• Amazon Currency Converter
• Your Account
• Your Orders
• Shipping Rates & Policies
• Amazon Prime
• Returns & Replacements
• Manage Your Content and Devices
• Recalls and Product Safety Alerts
• Conditions of Use
• Privacy Notice
• Consumer Health Data Privacy Disclosure
• Your Ads Privacy Choices

• School Guide
• Mathematics
• Number System and Arithmetic
• Trigonometry
• Probability
• Mensuration
• Maths Formulas
• Class 8 Maths Notes
• Class 9 Maths Notes
• Class 10 Maths Notes
• Class 11 Maths Notes
• Class 12 Maths Notes
• Calculus | Differential and Integral Calculus
• Limits in Calculus
• Formal Definition of Limits
• Properties of Limits
• Indeterminate Forms
• Strategy in Finding Limits
• Determining Limits using Algebraic Manipulation
• Limits of Trigonometric Functions
• Limits by Direct Substitution
• Estimating Limits from Graphs
• Estimating Limits from Tables
• Sandwich Theorem

Continuity and Differentiability

• Algebra of Continuous Functions - Continuity and Differentiability | Class 12 Maths
• Continuity and Discontinuity in Calculus - Class 12 CBSE

Derivatives

• Differentiation Formulas
• Algebra of Derivative of Functions
• Product Rule Formula
• Quotient Rule | Formula, Proof and Examples
• Differentiability of a Function | Class 12 Maths

Derivatives of Various Functions

• Derivatives of Polynomial Functions
• Derivatives of Trigonometric Functions
• Derivatives of Inverse Functions
• Derivatives of Inverse Trigonometric Functions | Class 12 Maths
• Derivatives of Implicit Functions - Continuity and Differentiability | Class 12 Maths
• Implicit Differentiation
• Implicit differentiation - Advanced Examples
• Derivatives of Composite Functions
• Chain Rule Derivative - Theorem, Proof, Examples
• Derivative of Exponential Functions
• Advanced Differentiation
• Logarithmic Differentiation
• Proofs for the derivatives of eˣ and ln(x) - Advanced differentiation
• Derivative of Functions in Parametric Forms
• Disguised Derivatives - Advanced differentiation | Class 12 Maths
• Second Order Derivatives: Rules , Formula and Examples (Class 12 Maths)
• Higher Order Derivatives
• Application of Derivatives
• Mean Value Theorem
• Rolle's Mean Value Theorem
• Lagrange's Mean Value Theorem
• Rolle's Theorem and Lagrange's Mean Value Theorem
• Tangents and Normals
• Equation of Tangents and Normals
• Increasing and Decreasing Functions
• Increasing and Decreasing Intervals
• Concave Function
• Inflection Point
• Curve Sketching
• Absolute Minima and Maxima
• Relative Minima and Maxima
• Antiderivatives
• Differentiation and Integration Formula
• Functions Defined by Integrals

Indefinite Integration

• Indefinite Integrals
• Integration Formulas
• Integration by Substitution Method
• Integration by Partial Fractions
• Partial Fraction Expansion
• Integration by Parts
• Integration by U-substitution
• Reverse Chain Rule
• Trigonometric Substitution: Method, Formula and Solved Examples
• Integration of Trigonometric Functions

Definite Integration

• Definite Integral | Mathematics
• Computing Definite Integrals
• Fundamental Theorem of Calculus | Part 1, Part 2
• Finding Derivative with Fundamental Theorem of Calculus
• Evaluating Definite Integrals
• Properties of Definite Integrals
• Definite Integrals of Piecewise Functions
• Improper Integrals
• Riemann Sums
• Riemann Sums in Summation Notation
• Definite Integral as the Limit of a Riemann Sum
• Trapezoidal Rule

Application of Integration

• Area under Simple Curves
• Area Between Two Curves - Calculus
• Area between Polar Curves
• Area as Definite Integral

Differential Equation

• Differential Equations
• Exact Equations and Integrating Factors
• How to solve Differential Equations?
• Separable Differential Equations
• Exact Differential Equations
• Homogeneous Differential Equations

Vector Calculus in Maths

Vector Calculus in maths is a sub-division of Calculus that deals with the differentiation and integration of Vector Functions. We already know that Calculus is a branch of mathematics that deals with the rate of change of a function concerning another function. There are two major divisions of Calculus namely, Differential Calculus and Integral Calculus .

The branch of Differential Calculus deals with the process of finding derivatives or differentiation of functions while Integral Calculus deals with finding the antiderivative of a function whose derivative is given. In this article, we will learn in detail about Vector Calculus which is a lesser-known branch of calculus, and the basic formulas of Vector Calculus.

In this article, you are going to read everything about what is vector calculus in engineering mathematics, vector calculus formulas, vector analysis, etc.

Table of Content

What is Vector Calculus?

Operation in vector, divergence and curl, vector calculus formulas, vector calculus identities, vector calculus applications.

• Solved Examples

Vector Calculus is a branch of mathematics that deals with the operations of calculus i.e. differentiation and integration of vector field usually in a 3 Dimensional physical space also called Euclidean Space. The applicability of Vector calculus is extended to partial differentiation and multiple integration. Vector Field refers to a point in space that has magnitude and direction. These Vector Fields are nothing but Vector Functions. Vector calculus is also known as vector analysis.

The vector fields are the vector functions whose domain and range are not dimensionally related to each other. The branch of Vector Calculus corresponds to the multivariable calculus which deals with partial differentiation and multiple integration. This differentiation and integration of vector is done for a quantity in 3D physical space represented as R 3 . For n-dimensional space, it is represented as R n .

Read in Detail: Calculus in Maths

Vector Calculus Definition

Vector calculus, also known as vector analysis or vector differential calculus, is a branch of mathematics that deals with vector fields and the differentiation and integration of vector functions

Vector Calculus often called Vector Analysis deals with vector quantities i.e. the quantities that have both magnitude as well as direction. Since we know that Vector Calculus deals with differentiation and integration of functions, there are three types of integrals dealt with in Vector Calculus that are 

Line Integral

Surface integral, volume integral.

Let’s learn about these integrals in detail.

Line Integral in mathematics is the integration of a function along the line of the curve. The function can be a scalar or vector whose line integral is given by summing up the values of the field at all points on a curve weighted by some scalar function on the curve. Line Integral is also called Path Integral and is represented by Φ = ∫ L f. Line Integral has got its application in physics. For Example, Work Done by Force is along a path given as W = ∫ L F(s).ds because we know that work done is given as the product of force and distance covered.

Surface Integral in mathematics is the integration of a function along the whole region or space that is not flat. In Surface integral, the surfaces are assumed of small points hence, the integration is given by summing up all the small points present on the surface. The surface integral is equivalent to the double integration of a line integral. Surface Integral has got its application in Electromagnetism and many more branches of physics where the vector function is spread over the surface. Surface Integral is represented as ∬ s f(x,y)dA.

Learn more about Double Integral .

A volume integral, also known as a triple integral, is a mathematical concept used in calculus and vector calculus to calculate the volume of a three-dimensional region within a space. It is an extension of the concept of a definite integral in one dimension to three dimensions.

Mathematically, the volume integral of a scalar function f(x, y, z) over a region R in three-dimensional space is denoted as:

[Tex]\bold{\iiint_R f(x, y, z) \ dV} [/Tex] Where  dV represents an infinitesimal volume element, and  Integral is taken over region R .

The different operations performed with vector quantities are tabulated below with their notation and illustration.

Learn More, Dot and Cross Product of Vectors

Divergence and Curl are two important operators used in Vector Calculus. Divergence is a scalar operator which tells about the behaviour of a function towards or away from a point. Curl is a vector operator which tells about the behaviour of a function around a point. The vector operator is represented by ∇ which accounts for the partial differentiation of the vector field. The Vector Differential Operator (∇) also called Nabla is expressed as ∇ = ∂/∂x i + ∂/∂y j + ∂/∂z k.

Divergence of Vector

If a vector field is given by f(x,y,z) = f x i + f y j + f z k then its divergence is given by taking the scalar of the vector operator is given by

div(f) = ∇.f(x,y,z) = (∂/∂x i + ∂/∂y j + ∂/∂z k) · (f x i + f y j + f z k )

⇒ ∇.f(x,y,z) = ∂ x /∂x + ∂ y /∂y + ∂ z /∂z.

Curl of Vector

If a vector field is given by f(x,y,z) = f x i + f y j + f z k then its curl is given by taking the vector of the vector operator

∇ × f(x,y,z) = (∂/∂x i + ∂/∂y j + ∂/∂z k) ⨯ (f x i + f y j + f z k )

⇒ ∇ × f(x,y,z) =  [Tex]\begin{vmatrix} i & j & k \\ \partial /\partial x& \partial /\partial y & \partial /\partial z \\ f_{x} & f_{y} & f_{z} \\ \end{vmatrix} [/Tex]

⇒ ∇ × f(x,y,z) = (∂ z /∂y – ∂ y /∂z)i + (∂ x /∂z – ∂ z /∂x)j + (∂ y /∂x – ∂ x /∂y).

Gradient of Scalar

The gradient of a scalar field F is given by grad(F) or ∇ F. It gives the measurement of the rate and direction of a scalar-valued function. In the Cartesian system, the gradient of a scalar-valued function is given by

∇ F = (∂/∂x i + ∂/∂y j + ∂/∂z k)F = ∂/∂x i + ∂/∂y j + ∂/∂z k

For a vector field given as F(x,y,z) = p(x,y,z)i + q(x,y,z)j + r(x,y,z)k. The following formulas are given.

Fundamental Theorem of Line Integral

if F = ∇Φ and Curve C has A and B endpoints then its line integral is given as

∫ c F.dr = Φ(B) – Φ(A)

Circulation Curl Form

There are two theorems under Circulation Curl Form, namely the Green theorem and Stokes theorem.

Green Theorem: If D is the region bounded by curve C then, ∮ c F.dr = ∬ D (∂Q/∂x – ∂P/∂y)dA

Stoke’s Theorem: For a surface S bounded by curve C stokes theorem given by ∮ c F.dr = ∬ S (∇ ⨯ F)dS

Flux Divergence Theorem

The Flux Divergence Form of Green’s Theorem is given as ∬ D ∇. F dA = ∮ c F.n ds

The Flux Divergence Form of Stoke’s Theorem is given as ∭ D ∇. F dV = ∯ s F.n dσ

The list of Vector Calculus Identitie s have been tabulated under three categories.

Gradient Function Identities

The Gradient Function Identities are tabulated below:

Divergence Function Identities

The identity formula for Divergence Function is tabulated below:

Curl Function Identities

The identities for curl function is tabulated below:

Laplacian Function Identities

The identities for Laplacian Function is tabulated below:

Degree Two Function Identities

The vector calculus identities for two degree function is tabulated below:

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

• 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) = 3xy 2 – y 2 z 3 then find gradF or ∇ f.

∇ f = (∂/∂x i + ∂/∂y j + ∂/∂z k)(3xy 2 – y 2 z 3 ) ⇒ ∇ f = ∂/∂x(3xy 2 – y 2 z 3 )i + ∂/∂y(3xy 2 – y 2 z 3 )j + ∂/∂z (3xy 2 – y 2 z 3 )k ⇒ ∇ f = 3y 2 i + (6xy – 2yz 3 ) + (-3y 2 z 2 )k

Example 2: Find the div(F) or ∇·F, if F = xz 2 i – 2y 2 z 3 j + x 2 yz k

div(F) = ∇·F = (∂/∂x i + ∂/∂y j + ∂/∂z k)·(xz2 i – 2y2z3 j + x2yz k) ⇒ ∇·F = ∂/∂x(xz2) + ∂/∂y(- 2y2z3) + ∂/∂z(x2yz) ⇒ ∇·F = z 2 – 4yz 3 + x 2 y

Example 3: Find curl F i.e. ∇ × f if F = xz 2 i – 2x 3 yz j + 2yz 4 k

∇ × f= (∂/∂x i + ∂/∂y j + ∂/∂z k) ⨯ (xz 2 i – 2x 3 yz j + 2yz 4 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(2yz 4 ) – ∂/∂z(-2x 3 yz)]i + [∂/∂z(xz 2 ) – ∂/∂x(2yz 4 )]j + [∂/∂x(-2x 3 yz) – ∂/∂y(xz 2 )]k ⇒ ∇ × f = [2z 4 + 2x 3 y]i + [2xz – 0]j + [-6x 2 yz – 0]k ⇒ ∇ × f = (2z 4 + 2x 3 y)i + (2xz)j -(6x 2 yz)k

FAQs on Vector Calculus

1. what is vector calculus.

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

2. What is Gradient in Vector Calculus?

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

3. What is Divergence in Vector Calculus?

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

4. What is Curl in Vector Calculus?

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

5. What is Line Integral?

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

6. What is Surface Integral?

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

7. Who invented Vector Calculus?

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

8. is Vector Calculus same as vector analysis?

Yes, Vector Calculus is also called vector analysis.

Please Login to comment...

Similar reads.

• Maths-Class-11
• Engineering Mathematics
• School Learning

Improve your Coding Skills with Practice

What kind of Experience do you want to share?