Calculus 3

9 Vectors in the Plane and in Space

9.1 Vectors in R2\mathbb{R}^2

Intro to Vectors

Cf. Vectors.

Vector Operations

Cf. Vectors: Vector operations.

The Unit Vector

Cf. Vectors: Unit vectors.

9.2 Coordinates and Vectors in R3\mathbb{R}^3

Three Dimensional Coordinate System

A “right handed” (if you stand at the origin with your right arm along the positive xx-axis and your left arm along the positive yy-axis, your head will point in the direction of the positive zz-axis) three-dimensional coordinate system has three perpendicular coordinate planes: the xyxy-, xzxz-, and yzyz-planes.

Equations of the coordinate planes in R3\mathbb{R}^3:

Plane Equation
yzyz x=0x = 0
xyxy z=0z = 0
xzxz y=0y = 0

In R3\mathbb{R}^3, the distance from the origin to (a,b,c)(a, b, c) is d=a2+b2+c2d = \sqrt{a^2 + b^2 + c^2}.

We can summarize the distance formula for P1(x1,y1,z1)P_1(x_1, y_1, z_1) and P2(x2,y2,z2)P_2(x_2, y_2, z_2)

P1P2=Δx2+Δy2+Δx2\lVert P_1 P_2 \rVert = \sqrt{ \Delta x^2 + \Delta y^2 + \Delta x^2 }

Graphs in R3\mathbb{R}^3

The graph of an equation in R3\mathbb{R}^3 is the collection of all points (x,y,z)(x, y, z) whose coordinates satisfy a given equation. This graph is called the surface.


To graph a plane, find some ordered triples that satisfy the equation. The best ones to use are those that fall on a coordinate axis (intercepts).


A sphere is defined as the collection of all points located a fixed distance (radius) from a fixed point (center).

Equation of a sphere: The graph of the equation

(xa)2+(yb)2+(zc)2=r2(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2

is a sphere with center (a,b,c)(a, b, c) and radius rr. This is the standard form of the equation of a sphere.

Solve given equations by rearranging to standard form and completing the square in the variables that require it.

Warning: Pay close attention to the signs when deriving the center coordinates from the equation since the coordinates are subtracted in the equation.


A cylindrical surface is a surface traced by a line moving parallel to a given fixed line and intersecting a given curve. A cylinder is defined with a generating curve (directrix) and a generating line (directrix). Lines running parallel to the directrix are called the rulings.

The curve is always in the plane containing the two variables in the equation. The rulings are parallel to the axis of the missing variable: e.g y=x2y = x2 has a generating curve in the xyxy-plane and rulings that run parallel to the zz-axis. Any three variable equation missing one variable will be a cylindrical surface.

Vectors in R3\mathbb{R}^3

A vector in R3\mathbb{R}^3 is a directed line segment in space.

Representation and operations are analogous to the representation and operations defined in R2\mathbb{R}^2.

9.3 The Dot Product

Cf. Vectors: Vector Operations, Dot (scalar) product.

9.4 The Cross Product

Cf. Vectors: Vector Operations Cross (vector) product.

9.5 Lines in R3\mathbb{R}^3

Equations of Lines in R3\mathbb{R}^3

Vector Equation of a Line in R3\mathbb{R}^3
r(t)=r0+tv=x0,y0,z0+ta,b,c\vec{r}(t) = \vec{r}_0 + t\vec{v} = \langle x_0, y_0, z_0 \rangle + t \langle a, b, c \rangle

where r0\vec{r}_0 is a position vector (points to a point on the line) and v\vec{v} is a direction vector (a vector parallel to the vector).

Parametric Form of a Line in R3\mathbb{R}^3:

A line parallel to the vector v=a,b,cv = \langle a, b, c \rangle that passes through the point (x1,y1,z1)(x_1, y_1, z_1) is given by

x=x1+tay=y1+tbz=z1+tcx = x_1 + ta \\ y = y_1 + tb \\ z = z_1 + tc

for some number tt.

Symmetric Form of a Line in R3\mathbb{R}^3

The parameter tt can be eliminated to obtain the symmetric form of a line

xx1a=yy1b=zz1c\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}

Parametric Equations

Converting a parametric equation to a rectangular equation is called eliminating the parameter. General guidelines are:

  1. Solve for tt in one of the equations
  2. Substitute into the second equation
  3. Simplify
  • If the equation involve trigonometric functions, trigonometric identities may be needed
  • The domain may need to be adjusted based upon the original parametric equation
Determining if Two Lines are Parallel, Intersecting (or Perpendicular), or Skew
  1. Test for parallel. If not, then
  2. Test for intersecting
  3. If intersecting, then test for perpendicular
  4. If not intersecting, then is skew

For example, given 3+2t,4t,1+3t\langle 3 + 2t, 4 - t, 1 + 3t \rangle and 1+4s,32s,4+5s\langle 1 + 4s, 3 - 2s, 4 + 5s \rangle

Test for parallel: Take the ratio of each component’s direction numbers (coefficients on the parameters) and set them equal to each other. Does 24=12=35\frac{2}{4} = \frac{-1}{-2} = \frac{3}{5}? No, therefore the lines are not parallel.

Test for intersecting: Solve a system of simultaneous equations. If we can solve it, then the lines are intersecting. Solve

3+2t=1+4s4t=32s1+3t=4+5s3 + 2t = 1 + 4s \\[16pt] 4 - t = 3 - 2s \\[16pt] 1 + 3t = 4 + 5s

Test for perpendicular: If the dot product of the two lines is 00, then the lines are perpendicular.

Cf. How to determine if two lines are parallel, intersecting (or perpendicular specifically), or skew on Krista King Math

9.6 Planes in R3\mathbb{R}^3

Forms for the Equation of a Plane in R3\mathbb{R}^3

A common way to specify the direction of a plane is by means of a vector NN (called a normal to the plane) that is orthogonal to every vector in the plane. The point-normal form comes from determining the dot product of all the points in the plane with the normal vector. The standard form then comes from distributing and simplifying.

A plane with normal N=A,B,CN = \langle A, B, C \rangle containing the points (x0,y0,z0)(x_0, y_0, z_0) has the following equations

  • Point-normal form: A(xx0)+B(yy0)+C(zz0)=0A(x - x_0) + B(y - y_0) + C(z - z_0) = 0
  • Standard form: Ax+By+Cz+D=0Ax + By + Cz + D= 0
Determine a Line’s Orthogonality/Collinearity With a Plane

A line is parallel to a plane if the direction vector of the line is orthogonal to the normal vector of the plane. Remember that two vectors are orthogonal if their dot product is zero.

A line is perpendicular to a plane if its direction vector is a scalar multiple of the normal vector of the plane. Remember that the normal vector is already perpendicular to the plane, so both being parallel means that their directions lie along the same line in R3\mathbb{R}^3.

Determine the Equation of a Plane Using a Normal Vector…
Orthogonal to a Given Plane:

Use the normal NN of the given plane and the given point to construct the equation in point-normal form.

Containing Three Given Points:

Given the points PP, QQ, and RR, a normal NN to the required plane is orthogonal to the vectors PRPR and PQPQ and is, therefore, found by computing with the cross product.

N=PR×PQN = PR \times PQ

You can now find the equation of the plane using this normal vector and any point in the plane.

Parallel to the Intersection of Two Planes:

Because the required line is perpendicular to the normals N1N_1 and N2N_2 of the given planes, the aligned vector is found by computing with the cross product.

N1×N2N_1 \times N_2

You can now find the equation of the plane using this normal vector and any point in the plane.

Vector Methods of Measuring Distances in R3\mathbb{R}^3

Distance from a Point to a Plane

The distance from the point P(x0,y0,z0)P(x_0, y_0, z_0) to the plane Ax+By+Cz+D=0Ax + By + Cz + D = 0 is given by

d=projnQP=QPNN=Ax0+By0+Cz0+DA2+B2+C2d = \lVert \text{proj}_n QP \rVert = \frac{| QP \cdot N|}{\lVert N \rVert} = \frac{| Ax_0 + By_0 + Cz_0 + D |}{\sqrt{A^2 + B^2 + C^2}}

Where QQ is any point in the given plane and NN is a normal to the given plane.

Note that to determine QPQP, we will need to determine one point on the plane QQ.

Obtain the Equation for a Sphere Tangent to a Given Plane

Given C(x0,y0,z0)C(x_0, y_0, z_0) and a plane, the radius rr is the distance from the center CC to the given plane (use the distance formula from above). Therefore the equation of the sphere is

(xx0)2+(yy0)2+(zz0)2=r2(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2
Distance from a Point to a Line

The distance from point %P% to the line LL is given by

d=v×QPvd = \frac{\lVert v \times QP \rVert}{\lVert v \rVert}

where vv is a vector parallel to LL and QQ is any point on LL.

Note that to determine QPQP, we will need to determine one point on the plane QQ (set t=0t = 0).

9.7 Quadric Surfaces

A quadric surface is given by a degree two equation in the general form

Ax2+By2+Cz2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0

where A,,JA, \ldots , J are constants.

Graphing Quadric Surfaces Using Traces

To graph a quadric surface, it is often helpful to graph the xyxy-trace, xzxz-trace, and yzyz-trace (the intersections of the surface with these three planes). To determine the xyxy-trace, sex z=0z = 0. To determine the xzxz-trace, set y=0y = 0. To determine the yzyz-trace, set x=0x = 0.

Common Quadric Surfaces


The general equation of an ellipsoid is

x2a2+y2b2+z2c2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1

If a=b=ca = b = c then we will have a sphere.


The general equation of a cylinder is

x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

If a=ba = b we have a cylinder whose cross section is a circle. The cylinder will be centered on the axis corresponding to the variable that does not appear in the equation.

Hyperboloid of one sheet

The general equation of a hyperboloid of one sheet is

x2a2+y2b2z2c2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1

The variable with the negative in front of it will give the axis along which the graph is centered.

Hyperboloid of two sheets

The general equation of a hyperboloid of two sheets is

x2a2y2b2+z2c2=1-\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1

The variable with the positive in front of it will give the axis along which the graph is centered.

Elliptic cone

The general equation of an elliptic cone is

x2a2+y2b2=z2c2\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}

The variable that sits by itself on one side of the equal sign will determine the axis that the cone opens up along.

Elliptic paraboloid

The general equation of an elliptic paraboloid is

x2a2+y2b2=zc\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z}{c}

The variable that isn’t squared determines the axis upon which the paraboloid opens up. The sign of cc will determine the direction that the paraboloid opens. If cc is positive then it opens up and if cc is negative then it opens down.

Hyperbolic paraboloid

The general equation of a hyperbolic paraboloid is

x2a2y2b2=zc\frac{x^2}{a^2} - \frac{y^2}{b^2} = \frac{z}{c}

These graphs are vaguely saddle shaped and as with the elliptic paraboloid the sign of cc will determine the direction in which the surface opens.

10 Vector-valued functions

10.1 Introduction to Vector Functions

Vector-valued Functions

A vector-valued function (or vector function) is a function where the domain is a subset of the real numbers and the range is a vector. It is given by

r(t)=f(t)i^+g(t)j^r(t)=f(t),g(t)r(t) = f(t)\hat{i} + g(t)\hat{j} \\[16pt] r(t) = \langle f(t), g(t) \rangle

in R2\mathbb{R}^2.

r(t)=f(t)i^+g(t)j^+h(t)k^r(t)=f(t),g(t),h(t)r(t) = f(t)\hat{i} + g(t)\hat{j} + h(t)\hat{k} \\[16pt] r(t) = \langle f(t), g(t), h(t) \rangle

in R3\mathbb{R}^3.

  • The domain of a vector valued function is the intersection of the domain of its components.

Operations with Vector Functions

Let FF and GG be vector functions of the real variable tt, and let f(t)f(t) be a scalar function.

Operation Equation
Addition (F+G)(t)=F(t)+G(t)(F + G)(t) = F(t) + G(t)
Subtraction (FG)(t)=F(t)G(t)(F - G)(t) = F(t) - G(t)
Scalar multiplication (fF)(t)=f(t)F(t)(fF)(t) = f(t)F(t)
Cross product (F×G)(t)=F(t)×G(t)(F \times G)(t) = F(t) \times G(t)

Limits and Continuity of Vector Functions

Limit of a vector function

Suppose the components of the vector function F(t)=ai^+bj^+ck^F(t) = a\hat{i} + b\hat{j} + c\hat{k} all have finite limits as tt0t \rightarrow t_0, where t0t_0 is any number or \infin or -\infin. Then

limtt0f(t)=[limtt0a(t)]i+[limtt0b(t)]j+[limtt0c(t)]k\lim\limits_{t \to t_0} f(t) = \left[ \lim\limits_{t \to t_0} a(t) \right]i + \left[ \lim\limits_{t \to t_0} b(t) \right]j + \left[ \lim\limits_{t \to t_0} c(t) \right]k
Rules for Vector Limits

Let FF and GG be vector functions of the real variable tt, and let h(t)h(t) be a scalar function so that all three functions have finite limits as tt0t \rightarrow t_0, then

Limit of a… Equation
Sum limtt0[F(t)+G(t)]=limtt0F(t)+limtt0G(t)\lim\limits_{t \to t_0} [F(t) + G(t)] = \lim\limits_{t \to t_0} F(t) + \lim\limits_{t \to t_0} G(t)
Difference limtt0[F(t)G(t)]=limtt0F(t)limtt0G(t)\lim\limits_{t \to t_0} [F(t) - G(t)] = \lim\limits_{t \to t_0} F(t) - \lim\limits_{t \to t_0} G(t)
Scalar multiple limtt0[h(t)G(t)]=[limtt0h(t)][limtt0G(t)]\lim\limits_{t \to t_0} [h(t)G(t)] = \left[ \lim\limits_{t \to t_0} h(t) \right] \left[ \lim\limits_{t \to t_0} G(t) \right]
Dot product limtt0[F(t)G(t)]=[limtt0F(t)][limtt0G(t)]\lim\limits_{t \to t_0} [F(t) \cdot G(t)] = \left[ \lim\limits_{t \to t_0} F(t) \right] \cdot \left[ \lim\limits_{t \to t_0} G(t) \right]
Cross product limtt0[F(t)×G(t)]=[limtt0F(t)]×[limtt0G(t)]\lim\limits_{t \to t_0} [F(t) \times G(t)] = \left[ \lim\limits_{t \to t_0} F(t) \right] \times \left[ \lim\limits_{t \to t_0} G(t) \right]
Continuity of a vector function

A vector function F(t)F(t) is said to be continuous at t0t_0 if t0t_0 is in the domain of FF and limtt0F(t)=F(t0)\lim\limits_{t \to t_0} F(t) = F(t_0)

10.2 Differentiation and Integration of Vector Functions

Vector Derivatives

The derivative of the vector function FF is the vector function FF^{\prime} determined by the limit.

F(t)=limΔt0ΔFΔtF^{\prime}(t) = \lim\limits_{\Delta t \to 0} \frac{\Delta F}{\Delta t}

wherever the limit exists. We say that the vector function FF is differentiable at t=t0t = t_0 if F(t)F^{\prime}(t) is defined at t0t_0.

Derivative of a Vector Function

The vector function F(t)F(t) is differentiable whenever the components functions are each differentiable.

F=f1(t)i+f2(t)j+f3(t)kF^{\prime} = f_{1}^{\prime}(t)i + f_{2}^{\prime}(t)j +f_{3}^{\prime}(t)k

Tangent Vectors

The vector derivative can be used to find tangent vectors to curves in space.

Smooth Curve

A vector function FF is smooth on a given interval if F(t)0F^{\prime}(t) \neq 0.

For example, if F(t)=a,b,cF(t) = \langle a, b, c \rangle, then F(t)0,0,0F^{\prime}(t) \neq \langle 0, 0, 0 \rangle.

Properties of Vector Derivatives

Higher-order derivatives of a vector function FF are obtained by successively differentiating the components of F(t)F(t). For example, the second derivative of FF is the derivative of F(t)F^{\prime}(t), and so on.

Modeling the Motion of an Object in R3\mathbb{R}^3

An object that moves in such a way that its position at time tt is given by the vector function R(t)R(t) is said to have

  • the position vector: R(t)R(t)
  • the velocity: V=R(t)V = R^{\prime}(t)

At any time tt,

  • the speed is the magnitude of the velocity: V\lVert V \rVert
  • the direction of motion is the unit vector: VV\frac{V}{\lVert V \rVert}
  • the acceleration vector is the derivative of the velocity: A=V(t)=R(t)A = V^{\prime}(t) = R^{\prime\prime}(t)

Vector Integrals

Vector integration is performed per component.

The indefinite integral of F(t)F(t) is the vector function

F(t)dt=[f1(t)dt]i+[f2(t)dt]j+[f3(t)dt]k+C\int F(t)dt = \left[ \int f_{1}(t)dt \right]i + \left[ \int f_{2}(t)dt \right]j + \left[ \int f_{3}(t)dt \right]k + C

where C=C1i+C2j+C3kC = C_1 i + C_2 j + C_3 k is an arbitrary constant vector.

The definite integral of F(t)F(t) is the vector

abF(t)dt=[abf1(t)dt]i+[abf2(t)dt]j+[abf3(t)dt]k\int_{a}^{b} F(t)dt = \left[ \int_{a}^{b} f_{1}(t)dt \right]i + \left[ \int_{a}^{b} f_{2}(t)dt \right]j + \left[ \int_{a}^{b} f_{3}(t)dt \right]k

10.4 Unit Tangent and Principal Unit Normal Vectors; Curvature

11 Partial differentiation

This chapter extends the methods of single-variable differential calculus to functions of two of more independent variables.

11.1 Functions of Several Variables

Basic Concepts

A function of two variables is a rule ff that assigns to each ordered pair (x,y)(x, y) in a set DD a unique number f(x,y)f(x, y). The set DD is called the domain and the corresponding values of f(x,y)f(x, y) constitute the range.

When dealing with such functions, we may write z=f(x,y)z = f(x, y) and refer to xx and yy as the independent variables and to zz as the dependent variable.

Level Curves and Surfaces

One way we can approach sketching the graph of a function of two variables, without the assistance of technology, is the same as graphing quadric surfaces in Section 9.7: Use the trace of of graph in a plane.

When the plane z=Cz = C intersects the surface z=f(x,y)z = f(x, y), the result is the trace, the equation f(x,y)=Cf(x, y) = C. The set of points (x,y)(x, y) in the xyxy-plane that satisfy this equation is called the level curve (or contour curve) of ff at CC. An entire family of level curves is generated as CC varies over the range of ff. Think of a trace as a “slice” of the surface at a particular location and a level curve as its projection onto the xyxy-plane.

Graphs of Functions of Two Variables

A more complete picture of a surface can be obtained by examining cross sections perpendicular to the other two principle directions as well.

If ff is a function of three variables xx, yy, and zz, then the solution set of the equation f(x,y,z)=Cf(x, y, z) = C is a region of R3\mathbb{R}^3 called a level surface of ff at CC.

11.2 Limits and Continuity

Single variable functions have domains that can be expressed in terms of intervals. However, functions of two or more variables requires special terminology and notation introduced in this section followed by limits and continuity.

Open and Closed Sets in R2\mathbb{R}^2 and R3\mathbb{R}^3

An open disk in R2\mathbb{R}^2 centered at point C(a,b)C(a, b) with radius rr is the set of all points P(x,y)P(x, y) such that (xa)2+(yb)2<r\sqrt{(x - a)^2 + (y - b)^2} < r. If the boundary of the disk is included, it is said to be a closed disk. Open and closed disks are analogous to open and closed intervals on a coordinate line.

A point P0P_0 is said to be an interior point of a set SS in R2\mathbb{R}^2 if some open disk centered at P0P_0 is contained entirely within SS. If SS is the empty set, or if every point of SS is an interior point, then SS is called an open set.

A point P0P_0 is said to be a boundary point of a set SS in R2\mathbb{R}^2 if every open disk centered at P0P_0 contains both points that belong to SS and points that do not. The collection of all boundary points of SS is called the boundary of SS. SS is said to be closed if it contains its boundary.

The empty set \emptyset is both open and closed in any topological space.

Similarly, an open ball centered at point C(a,b,c)C(a, b, c) in R3\mathbb{R}^3 is the set of all points P(x,y,zP(x, y, z) such that (xa)2+(yb)2+(zc)2<r\sqrt{(x - a)^2 + (y - b)^2 + (z - c)^2} < r. (etc. as above in R2\mathbb{R}^2).

Limit of a Function of Two Variables

For a function of two variables, the limit statement

lim(x,y)(x0,y0)f(x,y)=L\lim\limits_{(x, y) \to (x_0, y_0)} f(x, y) = L

means that for each given number ϵ>0\epsilon > 0, there exists a number δ>0\delta > 0 such that for all (x,y)(a,b)(x, y) \neq (a, b), if (x,y)(x, y) is in the open disk centered at (x0,y0)(x_0, y_0) in the xyxy-plane with radius δ\delta, then

f(x,y)L<ϵ| f(x, y) - L | < \epsilon

What this says is that the function value of f(x,y)f(x, y) must lie in the interval (Lϵ,L+ϵ)(L - \epsilon, L + \epsilon) whenever (x,y)(x, y) is a point in the domain of ff other than P0(x0,y0)P_0(x_0, y_0) that lies inside the disk of radius δ\delta centered at P0P_0.

If the limit is not the same for all approaches within the domain of ff, then the limit does not exist.


  1. Try direct substitution. If this results in an indeterminate form, then
  2. Check if the equation is factorable and cancel the factored terms. If not, then
  3. Take the limits (x,y)(0,0)(x, y) \rightarrow (0, 0) along the xx-axis and yy-axis or the lines y=xy = x and y=xy = -x.


Using the definition of the limit of a function of two variables, we can define the continuity of a function of two variables analogously. The function f(x,y)f(x, y) is continuous at the point (x0,y0)(x_0, y_0) if and only if

  1. f(x0,y0)f(x_0, y_0) is defined
  2. lim(x,y)(x0,y0)f(x,y)\lim\limits_{(x, y) \to (x_0, y_0)} f(x, y) exists
  3. lim(x,y)(x0,y0)f(x,y)=f(x0,y0)\lim\limits_{(x, y) \to (x_0, y_0)} f(x, y) = f(x_0, y_0)

The function ff is continuous on a set SS if it is continuous at each point in SS.

Limits and Continuity for Functions of Three Variables

The concepts introduced for functions of two variables in R2\mathbb{R}^2 extend naturally to functions of three variables in R2\mathbb{R}^2.

The limit statement

lim(x,y,z)(x0,y0,z0)f(x,y,z)=L\lim\limits_{(x, y, z) \to (x_0, y_0, z_0)} f(x, y, z) = L

means that for each number ϵ>0\epsilon > 0, there exists a number δ>0\delta > 0 such that

f(x,y,z)L<ϵ| f(x, y, z) - L | < \epsilon

whenever (x,y,z)(x, y, z) is a point in the domain of ff such that

0<(xx0)2+(yy0)2+(zz0)20 < \sqrt{(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2}

The function is continuous at the point P0(x0,y0,z0)P_0(x_0, y_0, z_0) if

  1. f(x0,y0,z0)f(x_0, y_0, z_0) is defined
  2. lim(x,y,z)(x0,y0,z0)f(x,y,z)\lim\limits_{(x, y, z) \to (x_0, y_0, z_0)} f(x, y, z) exists
  3. lim(x,y,z)(x0,y0,z0)f(x,y,z)=f(x0,y0,z0)\lim\limits_{(x, y, z) \to (x_0, y_0, z_0)} f(x, y, z) = f(x_0, y_0, z_0)

11.3 Partial Derivatives

Partial Differentiation

The process of differentiating a function of several variables with respect to one of its variables while keeping the other variable(s) fixed is called partial differentiation, and the resulting derivative is a partial derivative of the function.

If z=f(x,y)z = f(x, y), then the partial derivatives of ff with respect to xx and yy are the functions fxf_x and fyf_y, respectively, defined by

fx(x,y)=limΔx0f(x+Δx,y)f(x,y)Δxf_{x}(x, y) = \lim\limits_{\Delta x \to 0} \frac{ f(x + \Delta x, y) - f(x, y) }{\Delta x}


fy(x,y)=limΔy0f(x,y+Δy)f(x,y)Δyf_{y}(x, y) = \lim\limits_{\Delta y \to 0} \frac{ f(x, y + \Delta y) - f(x, y) }{\Delta y}

provided the limits exist.

This means that we find the partial derivative with respect to xx by regarding yy as a constant while differentiating the function with respect to xx. Similarly, the partial derivative with respect to yy is found by regarding xx as a constant while differentiating the function with respect to yy.

Alternate notation for partial derivatives for z=f(x,y)z = f(x, y)

fx(x,y)=fx=zx=xf(x,y)=zx=Dx(f)f_{x}(x, y) = \frac{\partial f}{\partial x} = \frac{\partial z}{\partial x} = \frac{\partial}{\partial x}f(x, y) = z_x = D_x(f)


fy(x,y)=fy=zy=yf(x,y)=zy=Dy(f)f_{y}(x, y) = \frac{\partial f}{\partial y} = \frac{\partial z}{\partial y} = \frac{\partial}{\partial y}f(x, y) = z_y = D_y(f)

Calculate the partial derivative of an implicitly defined function

zx=fxfz\frac{\partial z}{\partial x} = \frac{- f_x}{f_z}

where fx0f_x \neq 0.

Higher-Order Partial Derivatives

The partial derivative of a function is a function, so it is possible to take the partial derivative of a partial derivative. If we take two consecutive partial derivatives with respect to the same variable, the resulting derivative is called the second-order partial. We can also take a second partial derivative with respect to a different variable,producing what is called a mixed second-order partial derivative.

fxyfyxf_{xy} \equiv f_{yx}

11.4 Tangent Planes, Approximations, & Differentiability

11.5 Chain Rules

Chain Rule for One Independent Parameter

One independent variable tt.

dzdt=zxdxdt+zydydt\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}

Chain Rule for Two Independent Parameters

Two independent variables uu and vv.

dzdu=zxxu+zyyuanddzdv=zxxv+zyyv\frac{dz}{du} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial u} \\[16pt] \text{and} \\[16pt] \frac{dz}{dv} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial v}

12 Multiple integration

13 Vector analysis