Multivariable Calculus
This is a a collection of GeoGebra 3D applets that explain the main concepts in multivariable calculus.
Table of Contents
- Vectors and Curves
- Vectors in the Plane and Three dimensional Space
- Dot products and Cross products
- Lines and Planes in 3D Space
- Parametric Curves
- Calculus of Vector-valued Functions
- Motion in Space
- Length and Arc length Parametrization of Curves
- Curvature of Curves
- Functions of Several Variables
- Surfaces, Graphs, Level Curves, and Level Surfaces
- Limits and Continuity
- Partial Derivatives
- The Chain Rule
- Directional Derivatives and the Gradient
- Tangent Planes and Linear Approximation
- Maximum/Minimum Problems
- Multiple Integration
- Double Integrals over Rectangular Regions
- Double Integrals over General Regions
- Polar Coordinates
- Double Integrals in Polar Coordinates
- Triple Integrals
- Cylindrical and Spherical Coordinates
- Triple Integrals in Cylindrical and Spherical Coordinates
- Vector Calculus
- Vector Fields
- Line Integrals
- Conservative Vector Fields
- Green's Theorem
Vectors in the Plane and Three dimensional Space
[b][size=200][size=150]What is a Vector?[/size][/size][br][/b][br]Vectors are often used in physics to represent quantities like force and velocity when we need to specify both the magnitude and direction of those quantities. Usually, we represent a vector visually by an arrow. In the applet below, vectors in [math]\mathbb{R}^2[/math] and [math]\mathbb{R}^3[/math] are shown. [br][br][b]Remark[/b]: There is a very special vector that does not have any direction. What is it?
Vectors can be drawn as arrows pointing out from the origin (the point where all axes intersect). On a coordinate space (either 2D or 3D), the coordinates of the point at the arrowhead uniquely determine the vector. Therefore, for the vectors [math]\vec{v}[/math] in [math]\mathbb{R}^2[/math] and [math]\vec{u}[/math] in [math]\mathbb{R}^3[/math], we can express them mathematically as follows:[br][br][math]\vec{v} = \langle v_1 , v_2 \rangle[/math] and [math]\vec{u}= \langle u_1, u_2, u_3 \rangle[/math][br][br]where [math](v_1 , v_2)[/math] and [math](u_1, u_2, u_3)[/math] are points at the arrowhead of [math]\vec{v}[/math] and [math]\vec{u}[/math] i.e. P and Q respectively. [br][br][math]v_1[/math] and [math]v_2[/math] are called the [b]x-component[/b] and [b]y-component[/b] of [math]\vec{v}[/math] respectively.[br][br]Similarly, [math]u_1[/math], [math]u_2[/math], and [math]u_3[/math] are called the [b]x-component[/b], [b]y-component[/b], and [b]z-component[/b] of [math]\vec{u}[/math] respectively.[br]
(You can freely drag the points P and Q on the arrowheads of both vectors in the applet above to change the length and the direction of the vectors.)[br][br]Moreover, two vectors are regarded as equal if they have the same length and direction. Hence, the tail of a vector can be any point other than the origin i.e. we may shift the vector to another position if necessary, as long as the length and direction remain unchanged. Therefore, whenever you are given a vector in [math]\mathbb{R}^2[/math] or [math]\mathbb{R}^3[/math] and want to express it mathematically, you can just shift the vector to the origin and find the coordinates of the arrowhead.[br][br](You can freely drag the green vectors in the applet above to any position you like and they are considered as the same vectors as the black ones pointing out from the origin.)
[size=150][u][size=100]Postion Vectors[/size][/u][/size][br][br]As mentioned before, vectors are uniquely determined by the points at their arrowheads when pointing from the origin. Therefore, any point P (or Q) in [math]\mathbb{R}^2[/math] (or [math]\mathbb{R}^3[/math]) can be represented by the vector with its tail at origin and its head at the point P (or Q). It is called the [b]position vector[/b] of P (or Q). From this perspective, points and vectors are just two sides of the same coin. Sometimes it is convenient to regard points as position vectors.[br][br]
[u]Norm of a Vector[/u][br][br]How can we compute the magnitude (length) of a vector? Thanks to Pythoragas Theorem, we have the following definitions:[br][br]Given [math]\vec{v}=\langle v_1,v_2\rangle[/math] in [math]\mathbb{R}^2[/math], the [b]norm[/b] of the vector [math]\vec{v}[/math] is [br][br][math]|\vec{v}|=\sqrt{v_1^2+v_2^2}[/math][br][br]which is exactly the magnitude (length) of the vector [math]\vec{v}[/math].[br][br]For [math]\vec{u}=\langle u_1,u_2, u_3\rangle[/math] in [math]\mathbb{R}^3[/math], the [b]norm[/b] of the vector [math]\vec{u}[/math] is [br][br][math]|\vec{u}|=\sqrt{u_1^2+u_2^2+u_3^2}[/math][br][br]which is exactly the magnitude (length) of the vector [math]\vec{u}[/math].[br][br]By definition, the norm of zero vector is zero and the norm of any non-zero vector is positive.[br][br][br]The following applet show how the above formulas are derived from Pythoragas theorem.[br][br][br]
[b][size=150]Vector Addition[/size][/b][br][br]There are two main operations on vectors. The first one is the addition of two vectors. First, we consider two vectors in the plane, we can define their addition visually using the applet below:[br][br][list=1][*]Construct two vectors u and v in [math]\mathbb{R}^2[/math] using the vector tool [icon]/images/ggb/toolbar/mode_vector.png[/icon].[/*][*]Drag u to the origin i.e. the tail of u is a the origin.[/*][*]Drag v to the arrowhead of u.[/*][*]The vector u + v is defined as the vector pointing from the origin to the arrowhead of v.[/*][/list][br][br]This is the so-called [b]triangle rule[/b].[br][br]([i]Note[/i]: Here we write "u" and "v" instead of [math]\vec{u}[/math] and [math]\vec{v}[/math] for the sake of convenience.)
Alternatively, you can regard the vector u + v as the "diagonal vector" of the parallelogram formed by the two vectors u and v pointing out from the origin. This is the so-called [b]parallelogram rule[/b]. For the physics viewpoint, this definition of addition is quite natural. You can imagine two forces represented by u and v act on a mass at the origin. The resultant force is exactly u + v.[br][br]In the left pane of the above applet, when a vector u is created, its components [math]u=\langle u_1, u_2 \rangle[/math] are shown in the following format: [math]u=\begin{pmatrix} u_1 \\ u_2 \end{pmatrix}[/math]. Observe the components of the vectors when you add two vectors together and answer the following question:[br]
[u]Question[/u]: Given vectors [math]\vec{u}=\langle u_1,u_2\rangle[/math] and [math]\vec{v}=\langle v_1,v_2\rangle[/math], let [math]\vec{w}=\vec{u}+\vec{v}[/math]. What are the components of [math]\vec{w}=\langle w_1,w_2[br]\rangle[/math] ?[br]
The addition of two vectors in [math]\mathbb{R}^3[/math] are defined in the same way i.e. either using triangle rule or parallelogram rule. Moreover, suppose we are given two vectors [math]\vec{u}=\langle u_1,u_2,u_3 \rangle[/math] and [math]\vec{v}=\langle v_1,v_2,v_3 \rangle[/math] in [math]\mathbb{R}^3[/math],[br]
[u]Question[/u]: What are the components of [math]\vec{u}+\vec{v}[/math] ?[br]
[size=150][b]Vector Scaling[/b][/size][br][br]The second main operation on vectors is scaling. Suppose k is any real number and u be any vector in [math]\mathbb{R}^2[/math] or [math]\mathbb{R}^3[/math].[br][list][*]If k >0, then ku is the vector having the same direction as u such that its length is k times the length of u.[br][/*][*]If k = 0, then ku is a zero vector.[/*][*]If k < 0, then ku is the vector having the opposite direction to u such that its length is |k| times the length of u. (Note: |k| is the absolute value of k.). In other words, [math] |ku|=|k||u|[/math]. [/*][/list][br]In the applet below, you first construct a "slider" corresponding to the value of k by typing "k" and press Enter. Then construct a vector u and its scaling by k i.e. the vector ku. You can drag the slider to see how the vector ku changes for different values of k.[br]
[u]Question[/u]: Given a real number k and vector [math]\vec{v}[/math] in [math]\mathbb{R}^2[/math] and vector [math]\vec{u}[/math] in [math]\mathbb{R}^3[/math], what are the components of [math]k\vec{v}[/math] and [math]k\vec{u}[/math] ?[br]
[u]Unit Vectors[br][/u][br]A vector is called a [b]unit vector[/b] if its norm (length) equals 1. For any non-zero vector [math]\vec{v}[/math], we can scale it by factor [math] \frac{1}{|\vec{v}|}[/math] to unit vector in the same direction:[br][br]Let [math]\vec{u}=\frac{1}{|\vec{v}|} \vec{v}[/math]. Then [math]|\vec{u}|=\left|\frac{1}{|\vec{v}|} \vec{v}\right|=\frac{1}{|\vec{v}|}|\vec{v}|=1[/math]. (Note: Sometimes a unit vector can be written as [math]\frac{\vec{v}}{|\vec{v}|}[/math])[br][br]In [math]\mathbb{R}^2[/math], there are two special unit vectors [math]\vec{i}=\langle 1,0\rangle[/math] and [math]\vec{j}=\langle 0,1\rangle[/math] such that any vector [math]\vec{v}[/math] in [math]\mathbb{R}^2[/math] can be expressed in terms of [math]\vec{i}[/math] and [math]\vec{j}[/math] as follows:[br][br][math]\vec{v}=\langle v_1,v_2\rangle =v_1\langle 1,0\rangle+v_2\langle 0,1\rangle= v_1\vec{i}+v_2\vec{j}[/math].[br][br]Similarly, in [math]\mathbb{R}^3[/math], the three special unit vectors are [math]\vec{i}=\langle 1,0,0\rangle[/math], [math]\vec{j}=\langle 0,1,0\rangle[/math], and [math]\vec{k}=\langle 0,0,1\rangle[/math]. Moreover, any vector [math]\vec{u}[/math] in [math]\mathbb{R}^3[/math] can be expressed as follows:[br][br][math]\vec{u}=\langle u_1,u_2,u_3\rangle =u_1\langle 1,0,0\rangle+u_2\langle 0,1,0\rangle+u_3\langle 0,0,1\rangle= u_1\vec{i}+u_2\vec{j}+u_3\vec{k}[/math].[br][br][br][br][br]
[u]Vector Subtraction[/u][br][br]Vector subtraction can be easily defined in terms of addition and scaling as follows: [math]\vec{u} -\vec{v} = \vec{u} + (-1)\vec{v}[/math]. Also, the components of [math]\vec{u}-\vec{v}[/math] can be computed by doing the subtraction component-wise.[br][br]You can construct vectors [math]\vec{u}[/math] and [math]\vec{v}[/math] in the above applet and then find out [math]\vec{u} + (-1)\vec{v}[/math].[br][br]
[u]Question[/u]: Consider the parallelogram formed by two vectors [math]\vec{u}[/math] and [math]\vec{v}[/math], can you express its two "diagonal vectors" in terms of [math]\vec{u}[/math] and [math]\vec{v}[/math]?[br][br]
Suppose [math]A=(a_1,a_2)[/math] and [math]B=(b_1,b_2)[/math] are two points in [math]\mathbb{R}^2[/math]. The vector with [math]A[/math] as its tail and [math]B[/math] as its arrowhead is denoted by [math]\overrightarrow{AB}[/math]. As we know, [math]\langle a_1,a_2\rangle[/math] and [math]\langle b_1,b_2\rangle[/math] are the position vectors of [math]A[/math] and [math]B[/math] respectively. Therefore, we have[br][br][math]\overrightarrow{AB}=\langle b_1,b_2\rangle-\langle a_1,a_2\rangle=\langle b_1-a_1,b_2-a_2\rangle[/math][br][br]Since the norm of [math]\overrightarrow{AB}[/math] is exactly the distance [math]d[/math] between [math]A[/math] and [math]B[/math], we have[br][br][math]d=|\overrightarrow{AB}|=\sqrt{(b_1-a_1)^2+(b_2-a_2)^2}[/math], we have[br][br][br]Similarly, for any two points [math]P=(p_1,p_2,p_3)[/math] and [math]Q=(q_1,q_2,q_3)[/math] in [math]\mathbb{R}^3[/math], [br][br][math]\overrightarrow{PQ}=\langle q_1-p_1,q_2-p_2,q_3-p_3\rangle[/math][br][br]Moreover, the distance [math]d[/math] between [math]P[/math] and [math]Q[/math] is as follows:[br][br][math]d=|\overrightarrow{PQ}|=\sqrt{(q_1-p_1)^2+(q_2-p_2)^2+(q_3-p_3)^2}[/math][br]
[u]Properties of Vector Arithmetic[/u][br][br]Let [math]\vec{u},\vec{v},\vec{w}[/math] be vectors in [math]\mathbb{R}^2[/math] or [math]\mathbb{R}^3[/math] and [math]\lambda, \mu[/math] be real numbers.[br][br]1. [math]\vec{v}+\vec{u}=\vec{u}+\vec{v}[/math][br]2. [math](\vec{u}+\vec{v})+\vec{w}=\vec{u}+(\vec{v}+\vec{w})[/math][br]3. [math]\vec{u}+\vec{0}=\vec{u}=\vec{0}+\vec{u}[/math][br]4. [math]\vec{u}+(-\vec{u})=\vec{0}[/math][br]5. [math](\lambda\mu)\vec{u}=\lambda(\mu\vec{u})=\mu(\lambda\vec{u})[/math][br]6. [math]\lambda(\vec{u}+\vec{v})=\lambda\vec{u}+\lambda\vec{v}[/math][br]7. [math](\lambda+\mu)\vec{u}=\lambda\vec{u}+\mu\vec{u}[/math][br]8. [math]1\vec{u}=\vec{u}[/math][br]9. [math]|\lambda\vec{u}|=|\lambda||\vec{u}|[/math][br]10. [math] |\vec{u}+\vec{v}|\leq|\vec{u}|+|\vec{v}|[/math] (Triangle inequality)[br][br][br]
[u]Exercise[/u]: Let [math]\vec{a}=\langle 3,-1,0\rangle[/math], [math]\vec{b}=\langle 1,2,-2\rangle[/math], and [math]\vec{c}=\langle -1,4,5\rangle[/math]. Compute [math]3\vec{a}-4\vec{b}+2\vec{c}[/math].[br]
[u]Exercise[/u]: Let [math]A=(1,2,3)[/math] and [math]B=(10,20,15)[/math] be two points in [math]\mathbb{R}^3[/math]. Find the vector [math]\vec{v}=\langle v_1,v_2,v_3\rangle[/math] such that [math]\vec{v}[/math] has the same direction as [math]\overrightarrow{AB}[/math] and [math]|\vec{v}|=3[/math].[br]
Surfaces, Graphs, Level Curves, and Level Surfaces
[b][size=150]Some simple surfaces in 3D space[/size][/b][br][br]Besides planes, there are some simple surfaces in [math]\mathbb{R}^3[/math] that can be described by simple equations. [br][br][u]Cylinders[/u][br][br]Suppose [math]C[/math] is a closed curve on a plane. A [b]cylinder[/b] can be generated by projecting [math]C[/math] along the axis perpendicular to the plane. For example, if the closed curve [math]C[/math] is a unit circle on the xy-plane. Then the equation [br][br][math]x^2+y^2=1[/math][br][br]describe the cylinder of [math]C[/math] (a circular cylinder) parallel to z-axis because for any cross-section at [math]z[/math], [math](x,y,z)[/math] is on the cylinder as long as [math]x^2+y^2=1[/math] i.e. [math]x[/math] and [math]y[/math] are independent of [math]z[/math]. Hence [math]z[/math] does not appear in the equation.[br][br]In the applet below, draw the cylinder of the circle on the yz-plane centered at [math](0,0,0)[/math] with radius [math]2[/math] parallel to x-axis.[br][br][u]Spheres[/u][br][br]The equation for the sphere centered at [math](x_0,y_0,z_0)[/math] with radius [math]r>0[/math] is clearly as follows:[br][br][math](x-x_0)^2+(y-y_0)^2+(z-z_0)^2=r^2[/math][br][br]The above equation describes the fact that the distance between any point [math](x,y,z)[/math] and [math](x_0,y_0,z_0)[/math] equals [math]r[/math].[br][br]If we compress/stretch a sphere along the three axes by some factors, we will get a more general kind of surfaces called [b]ellipsoid[/b]. For example, you can draw the ellipsoid [math]x^2+\frac{y^2}{4}+\frac{z^2}9=1[/math] in the applet below.[br][br][u]Cones[/u][br][br]Consider the right circular cone with apex at the origin [math](0,0,0)[/math] such that it is open upwards and its semi-vertical angle is [math]\frac{\pi}4[/math]. The equation of the cone is as follows:[br][br][math]z=\sqrt{x^2+y^2}[/math][br][br]If we want to include the cone below the xy-plane, we can use the following equation:[br][br][math]z^2=x^2+y^2[/math][br][br]More generally, if the semi-vertical angle is [math]\theta[/math], the equation of the cone becomes[br][br][math]cz=\sqrt{x^2+y^2}[/math][br][br]where [math]c=\tan \theta[/math].[br][br]You can draw a cone in the applet below.[br]
[b][size=150]Multivariable functions[/size][/b][br][br]In this chapter, we mainly study [b]multivariable functions[/b] i.e. real-valued functions of more than one variable. Suppose [math]f[/math] is a real-valued function of two variables. We can express this function as follows:[br][br][math]z = f(x,y)[/math][br][br]where [math](x, y)[/math] is any point in the [b]domain[/b] [math]D[/math] of [math]f[/math], a subset of [math]\mathbb{R}^2[/math] containing all the possible inputs of the function. Then a unique real number [math]z[/math] is assigned by the function for each [math](x,y)[/math]. The [b]range[/b] of [math]f[/math] is the set of all possible outputs of the function i.e. the set of all possible values of [math]z[/math].[br][br][u]Example[/u]: Suppose [math]z=f(x,y)=\sqrt{4-x^2-y^2}[/math]. It domain is [math]\left\{(x,y) \ | \ x^2+y^2\leq 4\right\}[/math] and its range is [math]\left\{z \ | \ 0\leq z \leq 2\right\}[/math].[br][br]Similarly, we can also consider a function of three variables [math]w=f(x,y,z)[/math], or in general, a function of [math]n[/math] variables [math]y=f(x_1,x_2,\ldots, x_n)[/math]. [br][br]As we know, we can visualize a function of one variable [math]y=f(x)[/math] as a graph in [math]\mathbb{R}^2[/math] i.e. it is a set of all points of the form [math](x,f(x))[/math]. Similarly, we can visualize a function of two variable [math]z=f(x,y)[/math] as a graph in [math]\mathbb{R}^3[/math] i.e. it is a set of all points of the form [math](x,y,f(x,y))[/math].[br][br]In the applet below, you can draw the graph of [math]z=f(x,y)[/math] and then find the point [math]P[/math] on the graph corresponding to the point [math]A=(x,y)[/math].[br][br]You can try the following functions:[br][br][list][*][math]f(x,y)=1-x-\frac 12 y [/math][/*][br][*][math]h(x,y)=x^2+y^2 [/math][/*][br][*][math]g(x,y)=\sqrt{1+x^2+y^2} [/math][/*][br][/list]
[b][size=150]Level curves[/size][/b][br][br]Consider the graph of [math]z=f(x,y)[/math]. The [b]contour curve[/b] is the intersection of the graph and the horizontal plane [math]z=k[/math] for some real number [math]k[/math]. The [b]level curve[/b] is the projection of the contour curve onto xy-plane. It can be represented by the equation [math]f(x,y)=k[/math]. [br][br]In the applet below, the level curve [math]f(x,y)=k[/math] is shown in the left panel and the corresponding contour curve is shown in the right panel. You can view the level curves for the following functions:[br][list][*] [math]f(x,y)=\sqrt{x^2+y^2-1} [/math][/*][br][*] [math]g(x,y)=\frac yx [/math][/*][/list][br][br]Moreover, you can use the applet to draw multiple level curves (e.g. [math]f(x,y)=k[/math] for [math]k=0,1,2,3,4,5[/math] ) to form a [b]contour map[/b] of the graph using the following method:[br][br]First, enter the command "k=slider(0,5,1)". Then right-click on the level curve and check the box "Show trace". Drag the slider to generate the level curves for [math]k=0,1,2,3,4,5[/math].[br][br]
[b][size=150]Level surfaces[/size][/b][br][br]For functions of three variables [math]w=f(x,y,z)[/math], it is not easy to visualize its graph since it is a three-dimensional object in four-dimensional space. However, we can visualize the intersection between the graph and [math]w=w_0[/math] as it is a two-dimensional object i.e. a surface. It is called the [b]level surface[/b] of [math]f[/math]. Its equation is [math]f(x,y,z)=w_0[/math].[br][br][u]Example[/u]: Let [math]w=f(x,y,z)=x^2+y^2+z^2[/math]. For [math]w=w_0>0[/math], its level surface has the equation [math]x^2+y^2+z^2=w_0[/math] i.e. a sphere centered at the origin with radius [math]\sqrt{w_0}[/math].[br][br]In the applet below, you can visualize the level surface [math]f(x,y,z)=k[/math] for various functions.[br][br](Note: GeoGebra can only handle the plotting of level surface for quadratic polynomials in three variables.)[br]