Multivariable Calculus Cheatsheet
A cheatsheet of basic multivariable calculus.
Terminology and notation
Vector: We will denote vectors in the plane by \((x,y)\)
Orthogonal: Two vectors are orthogonal if their dot product is zero, i.e. \(\vec{v}=(v_1, v_2)\), and \(\vec{w}=(w_1, w_2)\) are orthogonal if \[ \vec{v}\cdot\vec{w}=(v_1\cdot v_2)\cdot(w_1\cdot w_2) = v_1w_1 + v_2w_2 = 0 \]
Composition: Composition of functions will be denoted \(f(g(z))\) or \(f\circ g(z)\).
Parametrized curves
We often use \(\gamma\) for parametrized curve, i.e. \[ \gamma(t) = (x(t), y(t)) \] The tangent vector \[ \gamma'(t)=(x'(t), y'(t)) \] is tangent to the curve at the point \((x(t), y(t))\). It's length \(|\gamma'(t)|\) is the instantaneous speed of the moving point.
Chain rule
For a function \(f(x,y)\) and a curve \(\gamma(t)=(x(t), y(t))\) the chain rule gives \[ \frac{d\,f(\gamma(t))}{dt}=\frac{\partial f}{\partial x}\bigg\rvert_{\gamma(t)} x'(t) + \frac{\partial f}{\partial y}\bigg\rvert_{\gamma(t)} y'(t) = \nabla f(\gamma(t))\cdot \gamma'(t) \]
Grad, curl and div
Gradient: For a function \(f(x,y)\), the gradient is defined as \(\text{grad }f=\nabla f=(f_x, f_y)\). A vector field \(\vec{F}\) which is the gradient of some function is called a gradient vector field.
Curl: For a vector in the plane \(\vec{F}(x,y)=(M(x,y), N(x,y))\), we define \[ \text{curl }\vec{F}=N_x-M_y \]
Divergence: The divergence of the vector field \(\vec{F}=(M, N)\) is \[ \text{div }\vec{F}=M_x + N_y \]
Level curves
The level curves of a function \(f(x,y)\) are the curves given by \(f(x,y)=\text{ constant}\). The gradient \(\nabla f\) is orthogonal to the level curves of \(f\).
Line integrals
The ingredients for line (also called path or contour) integrals are the following:
- A vector field \(\vec{F}=(M,N)\)
- A curve \(\gamma(t)=(x(t), y(t))\) defined for \(a\leq t \leq b\)
The the line integral of \(\vec{F}\) along \(\gamma\) is defined by \[ \int_\gamma\vec{F}\cdot d\,\vec{r}=\int_a^b\vec{F}(\gamma(t))\cdot\gamma'(t)dt=\int_\gamma Mdx+Ndy \]
Properties of line integrals
Independent of parametrization
Reverse direction on curve \(\Rightarrow\) change sign. That is, \[ \int_{-C}\vec{F}\cdot d\vec{r}=-\int_C\vec{F}\cdot d\vec{r} \]
Fundamental theorem for gradient fields
If \(\vec{F}=\nabla f\), then \(\int_\gamma \vec{F}\cdot d\vec{r}=f(P)-f(Q)\), where \(Q,P\) are the beginning and endpoints respectively of \(\gamma\).
Def: If a vector field \(\vec{F}\) is a gradient field, with \(\vec{F}=\nabla f\), then we call \(f\) a potential function for \(\vec{F}\).
Path independence and conservative functions
Def: For a vector field \(\vec{F}\), the line integral \(\int \vec{F}\cdot d\vec{r}\) is called path independent if, for any two points \(P\) and \(Q\), the line integral has the same value for every path between \(P\) and \(Q\).
Theorem: \(\int_C \vec{F}\cdot d\vec{r}\) is path independent is equivalent to \(\oint \vec{F}\cdot d\vec{r}=0\) for any closed path.
Def: A vector field with path independent line integrals, equivalently a field whose line integrals around any closed loop is 0 is called a conservative vector field.
Green's theorem
\(C\) is a simple closed curve, and \(R\) the interior of \(C\). \(C\) must be positively oriented (traversed so interior region \(R\) is on the left) and piecewise smooth (a few corners are okay).
Green theorem: If the vector field \(\vec{F}=(M,N)\) is defined and differentiable on \(R\) then \[ \oint_C Mdx+Ndy=\iint_R (N_x-M_y) dA \] In vector form this is written \[ \oint_C \vec{F}\cdot d\vec{r}=\iint_R\text{curl }\vec{F} d\,A \] where the curl is defined as \(\text{curl }\vec{F}=(N_x-M_y)\).
Extensions and applications of Green's theorem
Simply connected regions
Def: A region \(D\) in the plane is simply connected if it has "no holes".
Potential theorem
Potential Theorem: Take \(\vec{F}=(M,N)\) defined and differentiable on a region \(D\).
- If \(\vec{F}=\nabla f\) then \(\text{curl }\vec{F}=N_x-M_y=0\).
- If \(D\) is simply connected and \(\text{curl }\vec{F}=0\) on \(D\), then \(\vec{F}=\nabla f\) for some \(f\).
Extended Green's theorem
We can extend Green's theorem to a region \(R\) which has multiple boundary curves. Suppose \(R\) is the region between the two simple closed curves \(C_1\) and \(C_2\). Then we can extend Green's theorem to this setting by \[ \oint_{C_1}\vec{F}\cdot d\vec{r}+\oint_{C_2}\vec{F}\cdot d\vec{r}=\iint_R\text{curl }\vec{F}\,dA \]