Multivariable and vector calculus

Once a quantity depends on more than one input, single variable calculus is only the local tool. Height can depend on two map coordinates. Temperature in a room depends on three. A map can send a point in the plane to another point in the plane. Multivariable calculus extends derivatives and integrals to several inputs. Vector calculus organizes those ideas when the output, or the domain, lives in the plane or in space. This pillar answers whether the two names are the same subject, then maps who it helps, what the core pictures are, and which pages already on this site to open in order.

Are multivariable calculus and vector calculus the same?

They overlap heavily and are often taught in one course sequence, and departments use the names interchangeably for a third semester calculus course, which is why people ask whether vector calculus is the same as multivariable calculus. Multivariable calculus is the broader move from one input to several inputs: partial derivatives, multiple integrals, and constrained optimization. Vector calculus is the part that organizes fields, line integrals, surface integrals, divergence, curl, and Green's theorem, Stokes theorem, and the divergence theorem, when vectors live in the plane or in space.

Treat the catalog title as a label, not as a wall. A multivariable syllabus often still includes the vector theorems, and a vector calculus syllabus often still teaches partial derivatives, because divergence and curl are built from them. A few schools split the material across two terms. The calendar changes. The ideas still lock together.

The gradient is the bridge that makes the names feel interchangeable. It is a vector computed from a scalar function, so the technique is still multivariable calculus. The vector half starts when a field is given to you and you do not yet know that it is a gradient: line integrals, curl, and the theorems that move a derivative to the boundary.

Temperature on a map, wind over the map

Picture a weather map of a flat region. Temperature at each point is one number, say $T(x,y)=70+2x-y$, with $x$ measured east and $y$ measured north. That is a scalar function of several variables. One unit east raises the reading by $2$. One unit north lowers it by $1$. The gradient is the constant vector $(2,-1)$. It points toward hotter air. You built an arrow, but the map stored a number. That computation is multivariable calculus.

Wind on the same map is already an arrow. Let $\mathbf{F}(x,y)=(-y,x)$. At $(1,0)$ the arrow is $(0,1)$, so the air moves north. At $(0,1)$ the arrow is $(-1,0)$, so the air moves west. The arrows turn around the origin. Nothing in this field is a temperature. The object is a field of arrows. That is vector calculus.

The pictures meet, and they are not the same question. The gradient of temperature is a field you built from a scalar. This wind is a field the map gave you directly, and it is not the gradient of any temperature. The test uses the matrix of partial derivatives of $\mathbf{F}$, the Jacobian developed below. The columns are $(0,1)$ and $(-1,0)$, so the matrix is $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$. When the mixed partial derivatives agree, the Jacobian of a gradient is symmetric. The entries off the main diagonal here are $-1$ and $1$, which are not equal, so the matrix is not symmetric. The wind circulates. It does not come from a scalar temperature. The practice section checks the magnitude, the dot product, and the determinant after you write them.

Who this pillar is for

Start here if a course in several variables is in front of you, or if partial derivative symbols have appeared and the pictures have not. You should already be able to differentiate and integrate in one variable, and to say what a derivative measures. If that sentence is shaky, repair it on calculus foundations and what a derivative actually measures before you add inputs.

If matrices still feel like grids to grind, do the linear algebra first course first. Gradients, Jacobians, and divergence are much shorter once a matrix is a geometric machine. This page assumes you want the map of ideas. The pathway is the ordered route through the linear algebra pages that make the later steps possible.

Prerequisites

You need single variable limits (the epsilon delta definition), derivatives, and a picture of the definite integral as signed accumulation. You need coordinates in the plane. Dot products help immediately: a positive dot product means a similar direction, zero means a right angle (or a zero vector), and a negative dot product means an obtuse angle.

Read partial derivative, gradient, vector, and vector field when you want the short form of a word. OpenStax Calculus and MIT OpenCourseWare are on the sources page if you want a full development of line integrals and the big theorems. This pillar will not replace that book. It will tell you which idea to hold while you read it.

What you should be able to do

When you finish the core, you should be able to do these things in a sentence, then on a small example.

• Say whether multivariable calculus and vector calculus are the same subject, and name the main object of each.
• Compute a partial derivative by holding the other inputs fixed, and say which input was allowed to move.
• Build a gradient and say that it points toward steepest increase, perpendicular to a level curve.
• Write the linear approximation in several variables, and name a Jacobian as the matrix of that approximation when there are several outputs.
• Set up a constrained optimization as "gradients parallel," and know where to read the method: Lagrange multipliers intuition.
• Distinguish a scalar field, a vector field, divergence, and curl in words, and state what Stokes theorem equates.

What each course title usually covers

A course titled multivariable calculus usually moves from graphs of functions of several variables to partial derivatives, the chain rule, the gradient, tangent planes, and linear approximation. Maxima and minima follow, including constrained problems through Lagrange multipliers, then double and triple integrals. In many universities the vector theorems occupy the rest of the same term, still under that title.

A course titled vector calculus centers on vector fields: line integrals, the question of when a field is a gradient, Green's theorem, then flux, curl, divergence, Stokes theorem, and the divergence theorem. Some departments use that title for the whole third semester. Others use it only after a course that stopped at multiple integrals.

Read the outline, not the banner. If it lists both partial derivatives and Stokes theorem, this page maps the whole term. If it never mentions a vector field, save the Stokes reading. If it opens with line integrals, do not skip partial derivatives, because every component of a curl is one.

Partial derivatives and the gradient

A partial derivative holds every variable but one fixed and differentiates with respect to the free one. For $z=f(x,y)$, the symbol $\partial f/\partial x$ asks how $z$ changes when $x$ moves and $y$ stays put. It is an ordinary derivative along a line parallel to the $x$ axis. The other partial derivative is the ordinary derivative along a line parallel to the $y$ axis.

If the curve is given by an equation and you have not solved for one variable, you still want that slope. That is implicit differentiation. Implicit differentiation intuition writes the chain rule for a single input. Partial derivatives are the same idea once more than one input can move.

The gradient $\nabla f$ packages the first partials into one vector. Under the usual Euclidean geometry it points in the direction of steepest increase, and its length is that steepest slope. Walking along a level curve keeps $f$ constant, so the gradient is perpendicular to the level curve. Walking with the gradient changes $f$ fastest.

The gradient and directional derivative calculator estimates the same objects with a central difference. For $f(x,y)=x^{2}+y^{2}$ at $(1,2)$, the gradient is $(2,4)$, the magnitude is $\sqrt{20}$, and the directional derivative in the direction $(3,4)$ is $4.4$. The algebra on this page is still the authority. The calculator labels its output as an approximation.

Linear approximation and the Jacobian

Near a point where the partial derivatives behave well, a scalar function looks like its tangent plane:

$$f(\mathbf{x}) \approx f(\mathbf{a}) + \nabla f(\mathbf{a})\cdot(\mathbf{x}-\mathbf{a}).$$

That is the several variable cousin of $f(a)+f'(a)(x-a)$. The dot product with the gradient is the whole first order correction.

When the output is itself a vector, the derivative is a matrix. The Jacobian is the matrix of all first partial derivatives. Its columns say how the output moves when you nudge one input at a time. For a square map, the determinant of that matrix scales local area. A zero determinant means the approximation flattens area, so nearby outputs are squeezed onto a thinner set.

Read Jacobian matrices in plain language for the full story. A small check you can do before that page: let $F(x,y)=(x+y,\, xy)$. At $(1,1)$ every partial derivative equals $1$, so the Jacobian is $\begin{pmatrix}1&1\\1&1\end{pmatrix}$. The determinant is $0$. The two columns are the same vector, so the linear approximation collapses the plane onto the line through $(1,1)$. Confirm the determinant on the 2 by 2 matrix calculator after you have written the matrix yourself. That calculator multiplies and finds determinants. It does not compute partial derivatives. The Jacobian matrix calculator does, for formulas in x, y, and z.

Constrained optimization

An unconstrained local maximum in the interior occurs where the gradient is the zero vector. A constraint $g(x,y)=c$ leaves only the directions tangent to that curve. At a constrained max or min, $\nabla f$ is parallel to $\nabla g$, written $\nabla f = \lambda \nabla g$ when $\nabla g$ is not zero. The number $\lambda$ measures how the optimal value responds if you nudge the allowed level of $g$. Read Lagrange multipliers intuition once gradients are comfortable, with the Lagrange multiplier glossary line.

A geometry check, separate from the explainer's product example: the point on the line $x+y=1$ closest to the origin minimizes $f(x,y)=x^{2}+y^{2}$. Then $\nabla f=(2x,2y)$ and $\nabla g=(1,1)$ for $g=x+y$. Parallelism forces $x=y$, and the line forces $x=y=1/2$. The closest point is the place where the growing circle $x^{2}+y^{2}=r^{2}$ first touches the line, so the normals agree. That touch is the picture behind the algebra.

Vector fields, divergence, curl, and Stokes

A vector field assigns a vector to each point: wind, fluid velocity, or a force. A scalar field assigns a number. Writing "grad," "div," and "curl" in words is often clearer than a page of $\nabla$ symbols while you learn.

Divergence measures local expansion or compression. Curl measures local rotation. A line integral asks how much the field pushes along a path. A flux integral asks how much crosses a surface. Stokes theorem equates circulation around the oriented boundary of a surface with the flux of the curl through the surface. Green's theorem is the flat special case. Stokes theorem intuition works one example and explains why a kinder surface with the same rim gives the same circulation.

The vector operations calculator checks a dot product, a cross product, a magnitude, or a unit vector. A cross product's length is the area of the parallelogram the inputs span. That page does not compute a curl. For divergence and curl at a point, with each partial derivative shown and a picture of the arrows, use the divergence and curl calculator.

Suggested learning order

The naming answer above is the first checkpoint. Work in this order after it. If a step uses a word you cannot define, stop and read the glossary line before the next explainer.

1. If a derivative is still only a rule, refresh calculus foundations and what a derivative measures.
2. If matrices are new, follow the linear algebra first course through eigenvalues before the Jacobian.
3. Glossary: partial derivative and gradient. Compute $\nabla(x^{2}y)$ at $(1,2)$. You should get $(4,1)$. Then check $x^{2}+y^{2}$ at $(1,2)$ on the gradient and directional derivative calculator.
4. Jacobian matrices in plain language. Write a 2 by 2 Jacobian, compare it on the Jacobian matrix calculator, then check its determinant on the matrix calculator.
5. Lagrange multipliers with constraint. Do the closest point on $x+y=1$ as a second check.
6. Vector field, then Stokes theorem intuition. State the theorem in one sentence before the worked circle.
7. A full exercise set is on the sources page. This order is the cluster on this site.

Checkpoints and practice habits

Gradient check. For $f(x,y)=x^{2}+y^{2}$, the gradient is $(2x,2y)$, which points straight away from the origin. Level curves are circles. The radius is perpendicular to the circle. That is the gradient picture on the simplest hill. At $(1,2)$ the gradient is $(2,4)$. The gradient and directional derivative calculator should agree, and the directional derivative in the direction $(3,4)$ should be $4.4$.

Jacobian check. For $F(x,y)=(x+y,\, xy)$ at $(1,1)$, the Jacobian is $\begin{pmatrix}1&1\\1&1\end{pmatrix}$ and the determinant is $0$. Confirm the partials on the Jacobian matrix calculator, then the determinant on the matrix calculator after the matrix is on your paper.

Dot product check. The vectors $(4,1)$ and $(1,-4)$ have dot product $4\cdot 1 + 1\cdot(-4) = 0$, so they are perpendicular. Use the vector operations calculator only to confirm the zero. A zero dot product is the algebraic form of "this direction stays on the level curve to first order."

Wind check. At $(1,0)$ the field $\mathbf{F}(x,y)=(-y,x)$ equals $(0,1)$. The magnitude is $1$. The dot product with $(1,0)$ is $0$, so the arrow is perpendicular to the radius and the air is circling. The Jacobian is $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ and the determinant is $1$. Divergence is $0$ and the scalar curl is $2$. Confirm the magnitude and the dot product on the vector operations calculator, the determinant on the 2 by 2 matrix calculator, and the divergence and curl on the divergence and curl calculator, after the numbers are on your paper. Use the symmetry test from the weather section to decide whether the wind is a gradient.

How to know you understood

1. In two sentences, say what multivariable calculus takes as its main object and what vector calculus takes as its main object. A classmate should hear a number with several inputs, then a field of arrows.
2. From the weather picture, point to temperature as the scalar example and wind as the field example. Say which arrow you computed and which arrow the map already stored.
3. For $T(x,y)=70+2x-y$, state the gradient $(2,-1)$ and say that it points toward hotter air.
4. For $\mathbf{F}(x,y)=(-y,x)$, write the Jacobian $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ and say why the entries off the main diagonal show that the wind is not a gradient.
5. Name the next page for a Jacobian, for a constrained gradient, and for circulation: Jacobian matrices in plain language, Lagrange multipliers intuition, and Stokes theorem intuition.

Common pitfalls

Lagrange multipliers find candidates. You still compare values, and you still watch places where $\nabla g$ vanishes. A sign error in Stokes theorem is usually an orientation error. The Stokes explainer treats that with care.

A naming pitfall sits beside those. A double integral of a scalar density is still the scalar side, even in the plane or in space. Flux of a field is the vector side. Calling every gradient a reason to skip the field theorems mixes the two names again.

Related reading

The algebra under this page is linear algebra and its first course pathway. Single variable repair is calculus foundations. Proof habits are real analysis. Compare routes on the pathways hub.