= Solution
Use the positive <Laplace-Beltrami operator> on a compact <Riemannian manifold> without boundary. Its <heat kernel> is the smooth integral kernel $K(t,x,y)$, $t>0$, of the solution operators for the <heat equation>:
$$
(\partial_t+\Delta_x)K=0,\qquad
u(t,x)=\int_MK(t,x,y)f(y)\,dV_g(y),\qquad
u(t,\cdot)\longrightarrow f\quad(t\downarrow0).
$$
The last condition holds, in particular, in $L^2$ for smooth initial functions. The <heat trace> is
$$
Z_M(t)=\operatorname{Tr}(e^{-t\Delta})=\int_MK(t,x,x)\,dV_g(x).
$$
The <heat invariants> are the coefficients in the small-time <heat kernel expansion> of this trace:
$$
Z_M(t)\sim(4\pi t)^{-d/2}\sum_{j=0}^\infty a_jt^j,
\qquad a_0=\operatorname{Vol}(M),\quad
a_1=\frac16\int_M R_g\,dV_g.
$$
Equivalently, the local diagonal coefficients $u_j(x,x)$ integrate to $a_j$; $R_g$ is the <scalar curvature>. Some normalizations absorb $(4\pi)^{-d/2}$ into the coefficients. These definitions here use the displayed normalization consistently.
To prove uniqueness, let $K_1,K_2$ satisfy the <heat kernel> conditions and let $w$ be the difference of their solutions with the same smooth initial function. It solves $w_t=-\Delta w$ and tends to zero in $L^2$ as $t\downarrow0$. The <heat equation energy identity>, using <integration by parts> on the closed manifold, gives for $t>0$
$$
\frac{d}{dt}\|w(t)\|_{L^2}^2
=-2\langle\Delta w,w\rangle_{L^2}
=-2\int_M|dw|_g^2\,dV_g\leq0.
$$
Integrating from $s>0$ to $t$ gives $\|w(t)\|_{L^2}\leq\|w(s)\|_{L^2}$; sending $s\downarrow0$ proves $w(t)=0$. Thus for every smooth test function $f$,
$$
\int_M(K_1-K_2)(t,x,y)f(y)\,dV_g(y)=0.
$$
For fixed positive $t$ and $x$, smoothness makes this difference a smooth function of $y$. Taking its conjugate as test function shows its squared norm is zero. \b[The heat kernel is unique.]
The <compact elliptic spectral theorem> for the <self-adjoint> elliptic <Laplace-Beltrami operator> gives a complete orthonormal <basis> of smooth <eigenfunctions> $\phi_j$, with discrete <eigenvalues> $0\leq\lambda_0\leq\lambda_1\leq\cdots$ counted with multiplicity. Solving the <heat equation> for initial data $\phi_j$ gives $e^{-t\lambda_j}\phi_j$, so the unique <heat kernel> has its <eigenfunction expansion> and its <heat trace> is
$$
\boxed{Z_M(t)=\sum_j e^{-t\lambda_j}.}
$$
The series converges for every $t>0$. This proves that the <spectrum> with multiplicities determines the <heat trace>.
Conversely, its large-time behaviour recovers the distinct <eigenvalues> and their multiplicities successively. The multiplicity of zero is $m_0=\lim_{t\to\infty}Z_M(t)$. After subtracting the already determined terms, let
$$
R(t)=Z_M(t)-\sum_{\nu<\mu}m_\nu e^{-t\nu}
$$
be the positive remaining sum, and let $\mu$ be its smallest remaining <eigenvalue>. Then
$$
\boxed{\mu=-\lim_{t\to\infty}\frac{\log R(t)}t,\qquad
m_\mu=\lim_{t\to\infty}e^{t\mu}R(t).}
$$
Indeed, after factoring $e^{-t\mu}$, the next distinct <eigenvalue> is separated from $\mu$ by a positive gap. The remaining tail tends to zero: for fixed $t_0>0$ its absolute sum is bounded by a constant times $e^{-(\mu_{\mathrm{next}}-\mu)(t-t_0)}$, using finiteness of $Z_M(t_0)$. Induction recovers every spectral term. Thus \b[the heat trace and the spectrum with multiplicities determine one another].
Finally the leading coefficient among the <heat invariants> is positive, so
$$
\boxed{d=2\lim_{t\downarrow0}\frac{\log Z_M(t)}{\log(1/t)}.}
$$
The common <heat trace> of <isospectral manifolds> therefore forces them to have the same dimension. Once the dimension is recovered, its leading coefficient also recovers their common volume.
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