You CAN Ace Calculus

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

free ideas to save on bags & supplies

17calculus > vector fields

On this page we will give you an introduction to vector fields and how to draw them. We also have a few practice problems. Two main ways to work with vector fields involve the divergence and the curl.

### Difference Between Vector Functions, Vector-Valued Functions and Vector Fields

These three terms are easily confused and some books and instructors interchange them. In general, vector functions are parametric equations described as vectors. Vector fields usually define a vector to each point in the plane or in space to describe something like fluid flow, air flow and similar phenomenon. Vector-valued functions may refer to either vector functions or vector fields. Look carefully at the context and check with your instructor to make sure you understand what they are talking about.

In all three cases, you need to look at the context to see what is being discussed. To avoid confusion, we do not use the term vector-valued function on this site but some of the instructors in the videos we use refer to vector-valued functions.

This first video explains vector fields in detail, with lots of examples and graphs.

### Dr Chris Tisdell - Intro to vector fields [20mins-6secs]

video by Dr Chris Tisdell

As explained in the last video, you have already seen vector fields if you have learned how to calculate gradients since the gradient assigns a vector to each point in space.

Here is a second video explaining vector fields. He goes into more detail about applications and why it is important to have a correct understanding of them. There is some repetition but it is important to think about this from different angles to get a good perspective.

### Dr Chris Tisdell - What is a vector field? [42mins-46secs]

video by Dr Chris Tisdell

The videos above should be enough to explain the basics of vector fields. If you would like a couple of other perspectives, here are two more video clips explaining the same concepts.

### PatrickJMT - Vector Fields [1min]

video by PatrickJMT

### MIP4U - Vector Fields [4mins-16secs]

video by MIP4U

Okay, so that should be enough explanation to get you started on vector fields. You can find a few practice problems below. After that, the next topic discusses one way to manipulate vector fields, the curl.

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

Instructions - - Unless otherwise instructed, plot these vector fields.

$$\vec{F} = 2\hat{i} + 2\hat{j}$$

Problem Statement

Plot the vector field $$\vec{F} = 2\hat{i} + 2\hat{j}$$

Solution

### 827 solution video

video by Dr Chris Tisdell

$$\vec{F}(x,y) = -y\hat{i} + x\hat{j}$$

Problem Statement

Plot the vector field $$\vec{F}(x,y) = -y\hat{i} + x\hat{j}$$

Solution

### 828 solution video

video by PatrickJMT

$$\vec{F}(x,y) = -\hat{i} + \hat{j}$$

Problem Statement

Plot the vector field $$\vec{F}(x,y) = -\hat{i} + \hat{j}$$

Solution

### 829 solution video

video by MIP4U

$$\vec{F}(x,y) = -x \hat{j}$$

Problem Statement

Plot the vector field $$\vec{F}(x,y) = -x \hat{j}$$

Solution

### 830 solution video

video by MIP4U

$$\vec{F}(x,y) = x\hat{i} + y\hat{j}$$

Problem Statement

Plot the vector field $$\vec{F}(x,y) = x\hat{i} + y\hat{j}$$

Solution

video by MIP4U